When going to Goose Lake, WA is there any worsened chance that the car will not be able to "take" ?
Just about any car can handle a good gravel road, with the notable exception being low slung performance cars (cars with very little ground clearance). That being said, "gravel roads" vary a LOT. I have seen roads that were the next thing to pavement and gravel roads that had rock gullies which could and would take out your oil pan if you dropped a tire into them.
Looking at google maps, the approach appears to be a Forestry Road. While google has streetview for some forestry roads (which we could use to assess the quality), they don't for this one. In addition the condition of a gravel road can change rapidly due to bad weather. A road that was fine last year may be washes and ruts this year.
The only thing you can do it to go out and see in person. You should be fine, just use your best judgement on what YOU feel safe driving.
I can give you first hand experience with this. I have visited Goose Lake (and the Mount St Helens museum separately) over 5 times in the last 2 years. We drive a 2001 Dodge Grand Caravan with camping stuff and 3 kids in the back.
The road is very well maintained, with few potholes. I'd rank it (on roughness) just below a newly tarred and rocked side street. It IS narrow, and if trying to pass another car, you'll have to hug the side of the road. Think single-lane side street with cars parked on both sides narrow, in some places.
I drove around Namibia in 2011 in a 2000 Ford Focus, put nearly 8,000 miles on that car during that three-and-a-half week trip - and most of that trip was on sand or gravel roads. The car was fine at the end of the trip - a bit dusty, but perfectly serviceable and ran just great.
8 miles (16 in actuality if you ever intend to leave the campsite...) shouldn't pose any modern (post WW2) car any major difficulties it wasnt already going to have.
As with all other kinds of roads, it depends on its condition. The only universal concern with all gravel roads is the incidence of pieces of gravel getting thrown up by the tires and spoiling somebody's wind-shield or paint job.
The answer, of course, is to slow down and maintain proper following distance. 10-15 seconds behind the car in front of you gives plenty of time for any gravel to fall out of the air, and as long as your car's ground clearance is greater than the depth of the pot-holes you should be able traverse the road without damage if you go slowly enough. If it's a bad road, you might be limited to 5 MPH, in which case it would take you an hour and a half to get down the road, but it still beats the heck out of walking. :)
Watch out for mud, that can get nasty.
Take your winter chains and a bucket, shovel, and saw. If you're on Forest Service land, you're required to have the shovel anyway. (And possibly a few other things, I forget the exact list.)
A gravel road is not bad. We have a driveway that is gravel for half of its length, and I estimate we drove at least 275 miles on gravel during the last 5 years of our Volvo's life, when it was 12 to 17 years old. Our neighbors, who share the driveway, drove a similar distance in five years in an old, long, low Cadillac, and continue to do so
Most likely, an average car will be able to handle this road without a problem. Growing up, I was on many forest roads throughout Washington, often in a little old Honda Civic. Access roads for campgrounds and trailheads are usually in decent condition. If you get off the "main" forest roads (logging access), all bets are off and road conditions can be much worse.
Judging from Google maps, it looks like the campground sits on a "main" forest road, as I would expect. The "main" forest roads are maintained relatively well, with the occasional pothole or washboard section. Speeds of 20-30 mph are typical, and speed is usually limited more by curves and grade than poor road surface. Of course, my observations are generalities, and the current condition of any specific road can vary considerably.
The car could not care less what the material under its tires, as long as only the tires are touching the road. The main impact of driving on gravel is that the car will get dirtier than on a paved road. If you were accelerating fast enough, you would also notice that gravel is more slippery than a paved road.
The situation is different if the base of the car or some engine parts mounted near the bottom of the car are touching the road. In that situation the car could break down. Sometimes gravel roads have potholes, vegetation and other irregularities that could scratch the car base.
In the old world driving on gravel roads is a everyday thing. The cars are also inspected regularly for problems, and driving on gravel roads seems not to cause any specific problems to be found on these inspections.