Is your son a CUB Scout, working on the "Reading and Writing" Academics Belt Loop and/or Pin, or a BOY Scout, working on the Reading Merit Badge?
If he's a CUB Scout, then you can approve the work he's done.
If he's a BOY Scout, then the only person that can "sign off" on work toward the badge is his registered merit badge counselor. If that's the case, and when you said "His scout leader said his school teacher can sign off that he has done some of the requirements in class" you were referring to his merit badge counselor, then since you are his teacher, you can provide that verification.
Paul S. Wolf, P.E.(Ret.) mailto:firstname.lastname@example.org
Secretary, US Scouting Service Project, Inc.