I haven’t heard back yet so should I ask my mentor for an introduction?
Seems that it has been only 2 days since you applied this past Monday.
Thus, I suggest you wait a bit more before considering replying or doing follow-up. The "rule-of-thumb" that is often suggested on this Community is to wait at least a week before doing so, as doing it before could come as desperate or annoying to the recruiter or contact.
Furthermore, the fact that they are "connected via LinkedIn" doesn't mean they keep in touch, or have spoken recently; asking your mentor to introduce you may prove ineffective if this is true.
Before considering asking your mentor for such thing, first ask him/her if they are still in touch, to see if this idea is viable.