WARNING: ACTUALLY USEFUL ADVICE FOLLOWS
1. Is your plan feasible? Yes. However, you'll have to choose between making it all perfectly legal (business licenses, masseuse licenses, sales taxes, etc. everywhere you go, seeking permission from every place you stop, etc.) or working under the table/the radar. Personally, I feel that if you execute a private transaction with another individual for massage services, that's your and their business only. Stay unobtrusive and you might be able to just perform the service in their RV. Bottom line: your income will be sporadic. You might do quite well if you park in someplace like Quartzite. ALSO, set up a website, get a Twitter account, and constantly tweet your location. I think there might be a robust demand for your services if you market them properly. Mobile massage services. Sounds trendy.
2. I would NOT do what you're doing unless you had adequate provisions for child care, schooling, and medical care. Do you actually have the ability (not just the inclination) to home school your children? Will the oldest have the ability (and the inclination!) to care for your younger children while you're with clients?
3. Most important of all: are you contemplating doing this because you want to escape from your current situation? Are you finding motherhood and raising a family to be too constraining and confining? That's fine in and of itself. In fact, I would guess that fully half of the people our there who have started families never should have done so. Not everybody is cut out to be a parent. Our society is formed on the premise that everybody should get hooked up and start popping out babies as soon as possible, but I know quite a few people who met the ultimate departure of their children for the big wide world with a huge sigh of relief and the feeling that now, they could finally get out there and start LIVING. Implying, of course, that maintaining a household and attending to the little ones' every need and desire for decades, to them, hadn't really been living.
I suspect that may be the case with you. If so, going on the road fulltime won't eradicate that problem--you'll be putting it on your back and taking it with you. Of course, it might be seen as an elegant solution to the problem if you camped in locations with active grizzly bear populations and rubbed steak sauce into your kids' hair before they went to sleep at night. Over time, statistically, the problem would resolve itself.
If I have one overarching message, it's that what you are thinking about IS quite doable, but it will be complex. More and more people are living and working in a completely mobile fashion. If you can develop a following and market yourself properly, I see no reason why you shouldn't be able to at the very least, eke out a modest living. The attendant freedom, it sounds, could be well worth the hassle to you.