A few things......
First off, this hasn't been the best year for snow, either in terms of quality or amount! Oh well. :(
Cheap hostel, rentals, etc....... FORGET IT!! This is Jackson, land of paying $800 a month to have six slob roommates. You can find better rentals down in Alpine or over Teton Pass in Idaho. Which brings us to......
No 4WD=life sucks. Teton Pass/Snake River Canyon/Togwotee Pass get nasty.
Parking is not allowed overnight on any Jackson streets due to winter plowing.
Pretty much everyone in town--cops, private land owners, etc--sees a sprinter van (or any other van/camper) and knows what #vanlife is, so forget blending in.
If you don't have a shiny Mercedes Sprinter, i.e. a vintage Ford or Chevy camper, expect the cops and locals to be more wary of you camping in the open (especially true of Hippy VW vans).
If you do have a shiny new Sprinter van, you may or may not impress the other fake dirtbags (rent paying or #vanlife) that have driven the rest of us out. Point being the van isn't enough your personality matters. Even if yours has a custom bumper and snorkel, some other rich idiot will have spent more flash on theirs.
Wyoming cops in general are very much on the lookout for marijuana and the Jackson cops have been using dogs. Make sure all yours bulbs work! Certain license plates attract more attention.
On the other hand, thanks to the rich investors, trust funders, fake dirtbags and cops Jackson IS very safe so you won't have too much to worry about as far as van being broken into.
There is no free camping available on federal lands after the snow gets deep (at least nearby).
While winters haven't been nearly as cold on average, it still gets way below zero and has been much more damp.
There is a rec center for showering etc., but its moderately expensive.
Summer is a much better time to live in a van in Wyoming!!
Hope that helps.