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Lots of people do this.
In the RV world it's called "full timing".
Most full timers I've met have a large class A RV. You can do it in any RV though.
Campgrounds are everywhere that you can stay for a night, or months.
A lot of people on /out/ will tell you to not stay at campgrounds, as they're too expensive. They run $15- to $100 a night. From what I find $35-40 is the average.
You will need to visit them at minimum to dump your holding tanks. If you do a fast water saving shower, and are stingy with water use, you can go a week between dumping in a typical RV, much longer if you have large golding tanks. If you're washing dishes, taking long showers, and going the bathroom a lot, you may only get 2 days till you full the tanks.
In a car seems romantic, but it's a really shitty way to live. It's hard to sleep in, and hard to do just about everything (cook, shover, bathroom, ect). You can do it, and people do work with it, but it's not ideal.