I'm sure you can make it work but, I've never heard of anyone doing it. That doesn't mean it's not being done.
If you are definitely going to be stationary for a long enough period of time I don't see why you can't brew beer.
However, if you get the urge to move on or, CG neighbors become a problem, or there's a weather/natural disaster situation and you must go, that could be a problem.
Personally, I would brew enough at home and take it with me while traveling. If you are a full timer then my response is moot.
You will have a few people at campgrounds that see or smell something they don't understand. They may report you as a mobile meth lab. :)
And, if you're my neighbor at a CG, I'll be bugging you for a sample(s).
Good luck whatever you decide. Enjoy your travels.