Is it OK to apply moisturizer or aloe with lidocaine after beard transplant? When is the best time to start applying?

This is something that really should be answered by your surgeon.  My recommendation for my beard transplant patients is to keep the beard area absolutely dry for the first 5 days, after which normal but somewhat gentle face washing can proceed.

Once the crusts are off, you can do whatever makes sense to you. The use of oils or aloe is fine for skin care