Plexiglass mirror is available through some glass shops. It’s normally not kept in stock, so you might have to order it. You will need a special adhesive so the backing doesn’t get ruined.

Mirror Polished Stainless Steel.

A good glass shop will have access to Plexiglass mirror. They can order a piece to your size. Be aware that some adhesives will ruin the back of the Plexiglass so ask for their professional opinion on the correct adhesive.

By the time you find one the kid will have outgrown the vanity why does it need to unbreakable any mirror cut to size and glued in will work unless the kids throw this around it is fine.