I'm close by you. I got by 7MGE rebuilt to a 7MGTE by Toysport in Gardena at least 6 years ago. I'm neutral about their work. I couldn't really recommend them because the workmanship could be a lot better. They do specialize in Toyotas, but not any specific kind.
I've been looking at XS Engineering in Huntington Beach, or SP Engineering in the City of Industry. SP builds mostly Mk IVs, but I figure Mk IV is close to a Mk III. At least closer than a Honda. No one specializes in Mk IIIs that I know of, unless you want to go to Hollywood. XS does cars of any kind so they probably don't know Toyota Supras.
A conversion from a NA to T is kind of complicated. Not everyone can do it. You have to know the differences in the 7MGE and 7MGTE. If you're going to have a place do a conversion, SP would be the place to have it done, or of course do it yourself if you have the time and space.
As far as doing a conversion and California Smog laws, I don't know anything about the 1JZ so I can't say. But it is smog legal to do a 7MGTE conversion. But to make it real legal, at least the full GTE conversion, you'll need to get it smogged at the state referee for a waiver before you can start taking it to regular smog stations. The ignition systems are different. Or if you really know what you're doing, you'll get the placards changed to GTE ones and the guy doing the smog test won't know the difference. It would also be technically legal to use an aftermarket downpipe in place of the second cat. because the GE didn't have a second cat so you didn't alter or delete any smog equipment.
If you just want to turbo charge it (NA>1/2T) smog is not affected if you keep all the stock smog equipment on.