Lotta questions here...
So is it possible for a company or companies to design an RV that can withstand a nuclear bomb?
No, although this is mostly a question of distance from ground zero. To block a nuclear blast, one simply needs thick mass, which is incompatible with a vehicle that's designed to go places. That said, blocking a nuclear blast is also quite simple, you just need mass. Simply encourage people to park their RVs in ditches or specially dug underground garage pits and they will likely be fine since the blast/shock-wave simply passes overhead.
Sure. Nuclear fallout is not difficult to handle on an individual level, it is broadly speaking just irradiated dust. Any vehicle that's already designed with desert operation should be fine. Sealing the cabin to dust would be a bit more difficult, because you would need an integrated airlock/shower system instead of conventional doors, but again, totally doable.
and nuclear winter?
This one is a bit tricky because staying warm is a combination of preventing heat loss and generating more heat. Insulating such a vehicle well is doable, but at the end of the day, you are simply going to need large amounts of stored energy to make up for the energy loss (big fuel tanks).
While keeping everyone safe inside and still functional. What fuel does it need?
You need an energy dense fuel that holds long. This is because, realistically, the apocalypse RV is going to spend a lot of time sitting around being ready, but not going anywhere. Diesel holds a bit longer than Gasoline, however even with stabilizers, they are only going to hold 3 years top. Kerosene is probably be a better pick, because it is stable for up to 5 years standard and that's before adding fuel stabilizers and since it's basically heating oil, it should be salvageable from homes in an emergency.
How will survivors get clean water and air?
A reverse osmosis filter will clean water handily and they are quite simple and portable (although the do eventually require filter replacements) and can even treat wastewater. Water can be scavenged from basically anywhere and run through it into an onboard 'clean water' tank. Air filtration is even easier, since fallout-laden air is heavy, an air intake that simply faces downwards will already do a great job. Just slap a standard AC HEPA filter on it, and ensure that the RV is a positive-pressure environment, and you will be fine even if it isn't airtight.
How will they get food, if their rations run out?
They can simply buy/scavenge prepackaged food. Unless they are literally at ground zero where food has gotten irradiated directly, canned or otherwise airtight-packaged food that was made before the nuclear event will be fine. Fresh produce or other unpackaged food should be avoided though, since it may be contaminated with fallout.
Unless you are killed due to the immediate effects of a nuclear blast (shock/fire wave, collapsing building, radiation flash, etc) or die soon after the blast due to absorbing too much radiation, managing fallout isn't that difficult for an intelligent and prepared individual. The average home could easily be converted to be fallout proof with just a caulking gun, a HEPA filter, and an HVAC set to positive pressure. Similarly, your simple basic N95 mask would quite effective at dealing with fallout if you need to go outside (although a good shower afterwards is still recommended)
Hell, there are plenty of modern vehicles that are already "fallout proof", notably the Tesla Model S with the "bioweapon defense package" air filter system. All it takes is a decent filter, positive pressure, and you're golden.