It can be done, YES!
Lots of hard work.
I caution you on the 2 dogs, since you may be gone a lot working. Two dogs in a small RV with you gone working most of day is less than ideal.
Check out CoolWorks.Com for fun interesting jobs for energetic folks. Many include housing and are seasonal. Learn to save every penny you can NEVER go into debt. That is the REAL ticket to being young and following your dreams. Once you are saddled with debt, you can forget many dreams. Learn to deal with old and used. Then learn how to repair and maintain it. Numerous books on the subject, if you don't have a library card, then you aren't going to get far fast.
Read, read, read, all you can about RVing, learning new skills and so on.
Cooking is a great job that travels well, any where you go, they always needs cooks.
Keep to a minimalist lifestyle is the ticket to a low budget. I set out RVing on a budget and one of the things I gave up was changing to minimalist meat consumption in favor of dried beans. If it meant I could camp at the beach spring and fall, then I could eat beans year round. Life is a series of trade offs.
Check out Volunteer. Gov as they have many volunteer options with free RV parking. If you work fulltime and manage to volunteer for your camping spot, you can actually get ahead of the game financially.
For instance, Amazon is hiring right now for people with an RV and they provide endless overtime and free RV parking plus some perks.
GOOD LUCK.
Your dreams can work out if you are willing to work HARD and learn numerous new skills by reading lots of books.
People thought I was nuts when I ran off to the Caribbean to work on yachts, but it turned out to be a wonderful hard working career I enjoyed. I had to work from the ground up at it and soon I was in a beautiful position as a well paid crew member.
What you want to do is hard work and dedication but it can be done. You don't have to be a wealthy retiree to enter the wonderful world of RVing but slick money management skills are a must.
While some campgrounds require someone over 21, I've seen plenty that only require over 18.
A friend of mine who was young and loved camping and unconventional lifestyles, ended up in the Peace Corps for awhile, something else to consider.
There are also organic farms that are often looking for campers to work the farm and I know of two alpaca farmers who look for campers to work.
Also, find people that are impressed by you and get written character reference letters. These can be a huge help especially at your young age.
This is an excellent free workamping site:
http://www.work-for-rvers-and-campers.com/
GOOD LUCK and stick to the positive! Just because other think you can't doesn't make it true. If you think you CAN, then find a way to make it happen. Lots of wonderful opportunities come out of hard work and a willingness to lead an unconventional lifestyle.