I am a single parent myself and have been for 20 years. I'm 42 years old and have a 7 year old and a 20 year old. I purchased my motorhome last year and have had many fun times camping with both of my children, but am planning on taking a sabbatical (of 5-6 months duration) within the next six months with my 7 y.o. and a couple of furbabies. His 20 year old brother will stay at home.
I hadn't thought of hiring babysitters for my child while on the road, and don't think I will have occasion to. But if I were in your situation and I did want to work in an area, I would check out daycares in the area as many of them provide drop-in service. When your child is older, if you were staying in an area for a certain amount of time - particularly during school holidays, you could enrol them in a camp - for example a YMCA - usually they have to be over 5 years of age, but I believe the YMVA (where my son is currently at camp) accepts children at 3 years of age.
Here in Canada, many towns and cities have an organization which maintain a database of sitters or day home providers which they have vetted.
Obviously, you're taking a chance leaving your child with anyone, but I think that the current hyper-vigilant world of parenting today is giving society the perception that there are pedophiles and child abusers lurking behind every door.
To be frank, I think everyone who considers or lives this lifestyle is "escaping" in a sense, if only escaping a conventional or humdrum tick-tock life to live to the beat of their own drum.
I'm fortunate that I don't have to work while I'm traveling and that my son is young enough to homeschool for part of the year with relative ease. He also shares my sense of adventure, where his older brother always needed a greater sense of stability in his surroundings.
I see many people on this board and others who put off traveling until they were in their 60's and are quite physically impaired. What a shame that they had to wait that long to fulfil their dreams. Although I have great genetics with respect to my health situation, I also suddenly lost my mother when she was 56 and one of my sisters when she was 39. So if you can do it financially, I say go for it! Jump and the bridge will appear. Besides, nothing is final, you can always go back to your old life (perhaps with some modifications) and nothing is lost.
Personally, I would find it so hard to leave my 10 y.o. if I were you. What an experience traveling would be for a child of that age. It depends on the child though.
All the best of luck!

I love your tanacity Mama....I am a single gramma and also plan on camping with my grandaughter this winter for our first time.I will be homeschooling her and choose not to stay another gray/rainy winter here.I believe as you...dreams do come true...stay positive.We are heading for warmer weather and plan on having a good time while we are gone.You will do just fine with your 2 year old you have enough commen sense regarding her saftey...happy rving

Mama, I think what you are doing is fantastic . Home school the kids and take them to the places things happened. Just totally awesome. Babysitting will work out, it has too. Buying the rig, let us know, we will tell you if it is a good one or not. You may get a lot of opinions but the truth will bear fruit for you. Where I live at, some trailer parks and campgrounds there is always a "single" floating thru either working or doing what ever. The age group you are identifying with has a few years on you so you can expect to be parented, some bad, some good. Pretty safe stuff for a new lifestyle. See the movies RV and The Long Long Trailer
I went from Maine to New Mexico across country in a zig zag fashion with my new GF of the time and had the best possible time. Texans are friendly, a little weird at times, but friendly.
I envy your noble quest and I know you will make it work. Good luck...Phil

Congratulations Mama!
Everyone has given good advice and things for you to think about. They do this because they care and will, of course, be there for you down the road.
This life is about the journey. I have occasion to speak with people both young and old and my advice is always the same:
Children today really don't want things, they want love, protection, memories and a sense of identity from someone who cares enough to say no. They want someone that will love them and take them on the journey.

Take your children on the journey with you and thank God everyday for the time you each will have with the other. How many parents/children spend time saying, "I wished I would have........"? Too many IMHO.

By being diligent and thorough in your planning; you will go as prepared as you can. Life will "happen" wherever you are.

I hope we can meet sometime.