Gather a few plants from your area. For each one, flip a coin; if it's tails, you can use it for what you describe.
Where I'm coming from on this
I live near the area you specify; in New York, not in the Adirondacks but to the west of them, and I have been there. Fortunately though, my advice is generic.
I have just recently gotten deeper into the cord/rope making craft, and I have been pleasantly surprised by how easy it is to make and how many resources will work.
I have made a bunch of cords and ropes recently out of many different materials. The one part of your question that I cannot answer directly (but which I would say is not super important) is to be specific about species; I do not know what most of the plants around me are, but I just use them anyway, and most of them work! So here is my advice...
The Good Part
I have found that more plants seem to work than to not for this. Grasses, flowers, fibers, etc..
Anything that is soft and bendable is obviously a good first choice. Even if it is soft and breaks easily, that is usually fine if you just work in more of it.
One extreme of the spectrum: mowed grass
Even if it is short, that has not been a problem for me. Just a few days ago, I even decided to try it with short, mowed grass. I thought it would be difficult, maybe impossible, but I was surprised to find that even mowed grass, just 2 to 4 inches, made fine cord using the standard technique. I just started with a much smaller V to begin with the longest couple of grass blades I could find - I think those two blades were 4 or 5 inches - and I had to twist in extra material much more often than with longer resources. I made a cord that was a few millimeters thick, and it could have been used for some very lightweight purposes, and if made thicker it might have worked even for the table example you ask for.
Obviously, un-mowed grass that is a few feet long will work much better than mowed grass. This was just to make a point.
More plants will work if you soften them up first
Many plants that are not fibrous and which seem too tough and non-pliable to use, I have found that I can still use them if I soften them up first. For some of the plants, I carefully squeeze the stem until it partially crushes, and some still hold together decent even if fully crushed. This allows them to bend some and to tie around objects, especially if the objects are large such that the plant needn't form a tight bend.
Another softening technique
Similar to the previous technique, I have found that some plants become quite workable if I do the following, assuming they do not simply snap into two pieces when I try this but instead stays together after breaking: I bend the plant carefully until it breaks (but does not come apart; it is still 1 length), and then I either try to repeat this making many breaks along the plant, or if the plant is not too stiff sometimes I can "slide" the break (not sure how else to describe it) along the plant so that it is thoroughly broken along the entire length - some plants still hold together well after doing this. Then it is workable enough to use as with other methods.
Sometimes a processed cord is not necessary
Also, some plants seem to be strong enough to use directly, without processing into nice cord or rope. I have found multiple plants which I can directly pluck and straight away tie into a circle and have a difficult time breaking them; they would work great for the situation you describe. Sometimes these plants, though working without being cord or rope, do still require processing like in the previous two techniques I mention.
The default technique
Of course, there is the technique I see most advertised: try to get strands of fiber from plants, often from inside the bark of woody plants (including trees) and twist them together.
A tip on determination for the frustrated
Just because I make the bold claim that most of the plants around me work for tying/lashing, that does not mean that it's all sugar and spice.
What I do when I'm unsure of the resources in an area: I go for a short walk and take some time to look at all the different plants nearby, and I choose many of them that look promising and even some that do not look promising. I gather an armload of 5, or 10, or even more, different plant types - and often several plants of each type that I pick, sometimes it's more than an armload - and I sit down with them all and test each one.
This can take a while. Sometimes I spend an hour just experimenting with what I gathered, and I find that these 2 plants work ok, this 1 barely works, those 2 work terrible, this 1 simply cannot work at all, and this one works great. Then based on the best candidates, I decide which one I will use: which one was most abundant? easiest to pick? did it make me itch or sting when I handled it? etc.
Then I go gather. Depending on how far away the resource was or how plentiful or scarce it was: I will either gather several times more than I need to be safe (I still mess up and need to redo sometimes), or I gather just what I need and either go back for more if needed or sit and work at the resource's location.
The cliché phrase is very true here: if at first you don't succeed, try and try again. Do not be afraid to fail a lot.
Just this morning I tried several new (to me) resources for making baskets, and I insisted on limiting myself to the materials a family member had on their 1-acre property, even when they suggested some proper basket-weaving plants "just over their, off the property". I tried several times, and I did not complete any of the baskets I started; one of the materials was sort of working and maybe I could have finished a basket with a lot of very slow, careful, hard work, but I decided it was not worth it and moved on to some different materials. This morning's basket making attempts were very enlightening for me, but I would consider the basket making itself to have been a failure - even if I did finish the one that was sort-of working it would have been a weak and flimsy basket.
Another fun extreme: More grass
What inspired me to get back into this skill and develop it more was reading a website about making ropes from grass. You can find the site that re-inspired me here.
On that web page, a technique is discussed which uses multiple people and large bundles of grass to make large, strong ropes. The technique is based on the typical "take a bit of your material, make a V, keep twisting in material into each side" cordage technique, except it is scaled way up to a point where each person is doing what one or two of your fingers are doing when making the cordage by yourself.
The ropes that guy makes, with groups of people, are said to be thick but strong, generally taking loads of hundreds of pounds. He claims that one rope they made could not be broken in a game of tug-of-war between a large group of adults. And this is grass.
I am tempted now to include an entire section about some cultures which have (and still do in some cases) do neat things with grass ropes, such as the amazing bridges made out of grass ropes. But instead of delving into that, I will just mention it and suggest that Googling something like "ancient grass bridges" would provide interesting reading material. I would not want to trust a high bridge made of grass, but if done by skilled workers they do a fine job.
I used to have a hard time with this and considered paracord an integral part of my outdoor activities. But after experimenting with all of the plants I can find and practicing with them, I have found that it is actually difficult for me to go somewhere now and not find something that can be used for tying/lashing. Wherever I go, there seems to be multiple plants that fulfill the need. Even in the city, as long as it is a city with lawns, even fresh-mowed lawns provide adequate cord if you are determined enough - and I have done it from a city lot.
So, in essence, my answer is: The materials that work for your needs depends on your experience with them. Instead of researching which sources make the best cords or ropes, it is probably a better use of your time to learn how to make them from a larger range of sources. If you have practiced this, gather from 5 different plants (including trees) from the location you are in, and I would be surprised if you cannot find a suitable material for what you describe among what you have gathered. I cannot identify most of the stuff I use; I just use it and it just works.
Of course, I have not traveled extensively far away. Perhaps my thoughts about this would be dashed if I left the U.S. north-east and went camping in the mid-west or in a different country. But thinking back to videos I have seen of situations in other countries, I am fairly confident of my previous statements. Of course, an area that does not have multiple or abundant plants, like a desert, is another story entirely, but I think my answer should work for most of the world, and it is catered to New York state as you requested.
Also note: some of the materials may degrade quickly. I have never tested any of mine after a long time, but I have seen others say that their crude ropes (untreated, hastily constructed with poor weave or twist technique) has only lasted for months, or even days for some poor quality materials. I feel it important to add this so that you do not get the idea that all plants that work initially can be trusted to hold your large potting plant of garlic through the winter while you are away; when you come back in the spring that pot might be dumped on the ground. When I originally wrote this answer, I was envisioning something like "I'm roughing it for a week or two and I want my own crudely built stuff while I'm here so it can be 'luxurious roughing it'."