Can I use ear plugs to block out noise or will my ears need to pop because of the altitude? I don’t want to get sick by using the ear plugs.
Ear popping is due to the difference in the pressure between the outer and the middle ear. Popping of ear occurs at high altitude to allow for the two pressures to equalize.
Now, there's no correlation between ear popping and altitude sickness in the first place. So, in case you are worried about AMS due to the non-popping of ears, it wont happen. Second, as Sebastiaan has already pointed out, the plugs aren't fully sealed. They wont hold the pressure as you ascend while hiking. Hence, you will have to pop your ears even if you wear your plugs.
In short, the plugs wont harm you.
Earplugs are usually not 100% vacuum sealed. Even if you have custom-made ones that more or less seal (I have 2 pairs) they usually still lose the seal a bit occasionally. Certainly when you swallow, chew, yawn or do something different that moves something in your head. Combined with the extremely slow speed you’re ascending or descending (compared to an airplane or car) you really don’t have to worry about that.
The need for your ears to pop results from a pressure differential between the middle ear and the atmosphere. The middle ear is technically outside your body (e.g., the bones in the middle ear and both sides of the ear drum are covered in skin) because it is connected to the outside via the eustachian tube. The eustachian tube is usually closed so the pressure in the middle ear doesn't respond to changes in atmospheric pressure (e.g., due to changes in elevation or altitude). When the eustachian tube opens (for example when you clear your sinuses or chew) the pressure equalizes rapidly and you feel your ears pop.
If the external ear canal is perfectly plugged and then the atmospheric pressure changes there will be a small pocket of air with a different pressure than the middle ear once the eustachian tube open and equalizes the middle ear pressure with the atmospheric pressure. This pressure differential/force between the external ear and middle ear could cause pain and a feeling of fullness like your ears still need to pop. The exact impact will depend on the volume of your middle ear, the volume of the ear trapped behind the plug, and the pressure difference.
When an audiologist performs tympanography, they create an artificial increase in the pressure in the external ear. This is a sizable 10-20 Pa change in pressure, but still less than going from sea level to the top of Everest. The changes in elevation/altitude/pressure encountered while hiking/climbing are small compared to the pressure differences encountered in an unpressurized aircraft (aircraft can climb at well over 1000 ft per minute where as Alex Honnold and Dan Osman can only climb at 100 ft per minute), so if ear plugs are safe in unpressurized airplanes, they should be safe for climbing.
The FAA recommends ear plugs while flying. The military uses foam ear plugs (and sometimes ear plugs and muffs) in unpressurized air craft. The ear plugs do not form an air tight seal so therefore the outside pressure changes should not be a problem. If you experience discomfort like you ear not being able to pop, you can simply remove and then reinsert the plug.
If you are really concerned, or have experienced problems in the past (e.g., while flying), some companies sell plugs designed for flying (there may be other manufactures also and I have no experience with this product or relationship with the manufacturer).