Since that car is on Oregon plates, and you'll be registering it here in CA, you have a couple of cool options, rather than just getting handed a modern 7 digit plate.
1) CA allows YOM(Year Of Manufacture) license plates to be re-registered to your vintage car. You'll find these all day long on ebay, by searching "California Black and Gold License Plates" or "California 1966 License Plate." There are 2 constraints: a) You have to have both matching plates b) You have to have a rear-of-registration sticker on one of the plates, that matches the year model of the car. These original stickers are also available on ebay. Once you have the plates in hand AND have put the sticker on the plate that has been being used as the rear plate, you go to the local DMV and request the plates be registered to your car. They will take a photo of both plates, and send that, and your "request" paperwork to the division in Sacramento, for final approval. Once approved, you will get your correct registration documents, and a small metal plate. They want that metal plate to be mounted to one of the rear license plate attachment screws, and for you to put all future "current year registration stickers" onto that plate, so that the original sticker, in your case, a yellow "1966" sticker, is always displayed. IMO there's no need to do this. Just stick your new year stickers where they belong, on top of the 1966 sticker, and get on with your life.
2) CA offers a black and gold Heritage Plate, that you can custom number. So you could have a period looking set of plates, with some letter/number arrangement that you create, if it hasn't already been used......like on my Coupe, I could do something like "SEC 111" which would be a correctly configured number and letter set for the '60s, and be pretty cool. I found an even cooler set of plates, so I didn't do that.
Apologies for the unsolicited advice, and for writing a book!