Over the years since it's construction, it has been used for various science projects. It might contain NPS equipment or nothing. There are actually three rooms in the shelter, only one of which is open to the public.
Scattered around the high Sierra Nevada there are many snow-survey cabins, containing food, beds, stoves, etc. but the Whitney shelter not listed as a snow-survey cabin.
Here is an article about the shelter:
http://www.waymarking.com/waymarks/WMHQQV_Smithsonian_Institution_Shelter_Mt_Whitney_CAThen, scroll down and click on "NRHP Nomination form" to see an image of the original nomination form for the shelter, which describes the interior of the shelter.