Not quite. The word I'm looking for would be used to refer to a person who mirrors others in speech and action. If that definition of mirrorer is used enough, then perhaps an agent noun of mirror cannot exist. In any case, thanks for sharing it here!
I think you mean “mirrorizer” but it’s not an actual word (yet)! It’s from the verb mirrorize (obsolete; rare according to OED) and the ending -er (where the extra e is dropped).
It’s the same with other agent nouns that end with e for example:
hypnotize + er = hypnotizer
mesmerize + er = mesmerizer