Can You Have Mole Removal Before the Age of 18?

This should not be a problem. Moles can always form. It may not grow back from the same cells that are creating the moles you have. But you might have a predilection to forming moles and would just have to have future ones removed. But since you are under 18 you need to have your parent consent to the treatment.

Age has no bearing on mole removal but you should be old enough to understand what the procedure involves, the potential for scarring, and be able to tolerate the procedure.  If you wait, make sure to use sun protection.

I do not want to disagree with your mom, but in reality, this is not true. Moles do not "grow back" as long as they are removed correctly. However, based on the fact that you are under 18, you will not be able to have them removed without parental consent. So, if you mom insists you wait, then you're going to have to do this. But no, age has no bearing on whether or not a mole can be removed.