I've never noticed such service or lockers at BZE in many trips through it, but then I was never actually looking for it.
Since you're going to one of the islands -- I'm guessing Ambergris Caye -- you could make the trip to Flores while just leaving stuff at that hotel. That's the way I did it. Although in my case it was just a day trip; you'd probably want to negotiate not paying for the night in Belize when you're staying in Flores.
Enjoy Tikal -- it's impressive.