It depends. If you are able to talk to the delivery consultant before delivery, and maybe on delivery day, tell them that you will want to transfer tags and they may be able to do it, if the process has not already begun. If it has begun, then you should be able to trade, but that's up to the DMV and may have weeks of waiting before you can mount the old tag.
I believe that Tesla may use third parties, probably folks that do it for other dealers as well.
For my car in Georgia, the EV plate is a special one. Tesla normally registers the car for a standard plate. We talked to the delivery consultant and they gave us the option of either Tesla doing the normal registration and then we have to go back and get the EV tag, or we do the registration and then Tesla refunds us, since registration is part of the purchase price, we chose the latter and it worked out well for us, but that was indeed back in May. Interesting part is that the DMV knew the car existed, but didn't have it registered.
Honestly, I think that Tesla does about as equally poor in this department as every other auto dealer that I've dealt with. It takes way too long for something that should be close to instantaneous.