If eBay wants to serve that market, they'd need to meet the requirements regulating the teaching materials, in which case they'd need a system for certifying the bidders.
If eBay chooses not to, it is an opportunity for a third party - who will also need some way of certifying recipients.
I've homeschooled for seven in two states. In neither was there a requirement for anyone to prove they had the right to own a teacher's manual, nor should there be. In all my time on homeschool boards and in homeschool groups, I've never heard of having to "prove" eligibility to buy a teacher's manual. My county even gives old textbooks, including teacher's manuals, to anyone who wants them for free.
It's up to the parents of public and private school students to know what their kids are up to, not up to homeschool parents to prove a need. Once again non-parenting "parents" are taken completely off the hook while responsible parents are required to jump through hoops. Proving one has authorization to purchase a legal item is a slippery slope we as a society don't need to go any further down.
EBay, as a private entity, has the right to sell (or not sell) anything they choose for whatever reason they choose. But any individual should have the right to buy any legally available book somewhere without having to prove need.