Schools do accept kids over the summer. I was going to be home schooled at one point, and was accepted to a school over the summer. You need to use all the protektzia you can get.
Tell your neighbors, friends, Rabbi, etc. and G-d willing your children will be in a school by next year. Don’t give up. Dedicate time every day to calling people.
On the other hand, if you really want to home school, that is a valid choice. But you shouldn’t feel forced into it.