Well typewriter in the general shouldn’t cause any allergies. They are made mostly from metal/plastic and rubber. But this doesn’t mean you are safe near them (argh!).
First of all, and most obvious is dirt in any form, that settles on the machine like on any other thing in your room. But also it can be furr after pet of previous owner etc. Problem is, that typewriter have complicated mechanics, with lot of slots and places hard to reach during cleaning.
You can solve it in different ways:
Use plastic cover.
Clean it with compressed air from time to time.
Hoover it (watch for ribbon! It instantly is sucked in if you are not careful enough).
Most typewriters from unprofessional source need good cleaning before usage!
Another case is… the case.
I bought a lot of different typewriters and lot of their cases was dirty/stinky and covered in lot of different, hard to clean, stains.
I can imagine that some could have problem with mold or mites!
If you can’t clean it good enough (sometimes It is impossible, where there is a risk of damaging old, fatigued case) It’s better to keep it in different place (garage/attic etc.)
Ribbons - It’s not so obvious but this part is most dangerous. It’s wet, always on the top and can even be covered in mushroom in extreme cases. If you buy an old typewriter, and you don’t know how long ribbon was used - it’s best to throw it away.
Sound insulation - in many typewriters we have some material to silent at least some noises. It’s getting very dirty after years, since it’s structure absorbs all different things.
With my typewriters I make sure to clean it inside and outside in order to get rid of all allergens.
I’m writing as an allergic to dirt/mites, with allergic son (since he is still young, he react to even small amount of allergen in the air).