Can you do research in a federal wilderness area if it is non-invasive?

Question

i.e. snow depth measurement and sampling?

Answer 1

You probably don't even need permission to do snow depth measurement and sampling, so long as you are allowed to be in the area. Still, it would be courteous to let the people in charge of the area know what you are doing.

Answer 2

You need to find out who administers the particular wilderness area you want to work in and contact them.  Generally there is no problem if you prearrange things and don't permanently mount equipment.  The agency might be the Forest Service, BLM, Park Service or the Fish and Wildlife Service.