I am single and homeschool. It's not easy, but it's a priority of mine so I make it happen. And yes it's legal to be a single, homeschooling parent.
I get nothing from my dd's dad. I get some SSI for money, but not even close to being enough to cover our expenses. I also do a lot of babysitting (I bring dd with me). Sometimes I go-go dance or teach exercise classes for extra money. I live in a very nice place, but I also do a lot of heavy duty farm/barn work in addition to paying rent so that I can live here. To make things more fun I also go to college full time at night.
This year I enrolled dd in a homeschool program through a local charter school. They pay for whatever curriculum I want and they also provide free classes, so that may help me out some.
As far as general relief or welfare goes, you do need to be working or actively seeking employment to get benefits (in my state there are some exceptions for pregnancy and people caring for very little ones). You can get food stamps and medical though. But no, the government doesn't pay for single parents to stay home and homeschool. Won't your ex-husband be paying child support and helping out?