Something like a fiberglass egg; Scamp, Happier Camper, and many more might fit the bill. DW and I had a 16' scamp. Had a two burner stove, shower/toilet, two-way fridge (propane and AC), A/C was basically a window unit that I installed (or course you'd need to have a generator or shore power). We went cross country a few times pulling it with our Jeep Rubicon with no issues other than some sway which was cured by adding a sway bar to the hitch.
We only us my Vetran discounted T-mobile phone for Internet with great results and can stream to our smart TV, and computer pretty much everywhere. If was going to run a business, I'd go with a satellite service.
Regarding how long you can stay.. depends on the place - Find a farmer and do some chores for them... Public lands will restrict you to two weeks and require you to move from the general area before you come back. But, some folks will make multiple pseudo overlapping reservations and simply flit-back and forth for two weeks at a time which could fit with your need to re-supply.
Since you are young, energetic and probably physically capable you might look into workamping - where you might have to put in a few hours each day and get a site with water/electricity for free - but, you'll be dealing with campers and we can be a rowdy group (LOL).
