I don't recall, but I think he was a felon in possession, at least in earlier seasons. He didn't carry one around very often.
Coincidentally, I've currently been rewatching
The Rockford Files on Netflix. I'm just up to season 2 right now, but he's always getting in trouble for "not having a permit". He's been a fully pardoned person since episode 1, so could have gotten a carry permit, especially as a PI. Back then, there were parts of CA where you could get a permit in a few days with no tests or any of the other current state required crap.
It's been kind of funny watching the show today and actually thinking about what would happen if it was rebooted as is. I'm pretty sure Rockford would have been arrested or sued a bajillion times for every time he walks somewhere with a woman, as he always puts a hand on their arm (sex predator!) and does other stuff, common at the time, that would give current feminists heart attacks.
On a tangent, it has been kind of funny to watch Rockford do his job with the tech of the time (the show came out when I was 15, so I was around for that tech too). I keep thinking about how fast an episode would be over if only they had cell phones.
Though it's also funny to see how many of the bad guys had the clunky old car phones in their cars.
I also watched one episode based on trying to figure out what "5.56" was. Even though it had been used in the M-16 for I think ten years already, it was in the episode as a "new" military cartridge.
To this day I'm trying to figure out how he was able to take over a dozen parking spaces with his trailer on prime beach property in Malibu. It would be like $10,000 a month rent today.