You could certainly mail in the paperwork certified mail, nothing in state code about having to show up in person. Even if your county requies fingerprints, that can only be on *first time* applicants so don't let anyone try to convince you CHP renewal needs fingerprints too.
Problem might be in receiving your CHP since some counties require you
to come pick it up.
Moderator Mike, how would you force a locality to mail it to you, or release
it to a 3rd party? 45 day clock is for issuing the permit, does that imply
ensuring it is physically in your hands?
Also, if your renewal was denied then the clock starts (21 days?) for
requesting an ore tenus hearing.