will I be able to recoup the costs I spent for him to take a summer school class online, through my taxes this coming year?
There aren't any tax credits available for education expenses until after graduating from high school so Mr Placid is wrong.

The only thing I can think of is to see if the school district will reimburse you. I doubt it will unless it is a class that the school requires, but doesn't offer during the year.

I hope this helps.
Gary
Maybe, maybe not.

Start off by reading IRS forms 8863 and 8917.

Edit: OK, Gary is right. The tuition deduction is not available unless the student already has a high school diploma or GED. But from my reading (and Gary can correct me if I'm wrong) the education credit is available even if the student has not yet graduated high school, e.g. if the student, while in high school, takes a college course for college credit, and the college credit rolls over into a college degree program. But, in your son's case, I presume his online course was not for college credit, so no tax deductions or credits are available.
From the wording of your question, your child is pre-college. Tuition paid for a child is not deductible and there is no tax credit for this expense.

An exception is if your child needed to attend a special needs school, such as a school for the blind. In this case the cost of instruction would be deducted as a medical expense.
Assuming he's somewhere in the K-12 grade range, no. And even if it's college where it might be eligible for an education credit, you don't recoup the full cost, only part of it.