It depends. If the DMV has issued the sticker and registration within the last 30 days, then AAA will not be able to issue replacements. If they have issued it and it has been greater than 30 days, they are able to issue replacements. You will need to fill out a form (Reg 152) which they will have in the branch. Depending on your situation, you may have to pay a $21 replacement fee. AAA can only accept cash or check for any DMV payments. As long as you have your plate number or VIN, the will be able to see what is going on with the vehicle record. One more thing, DMV services are reserved only for AAA members. If you are not a member, an agent in the branch can get you a membership the same day so you can complete your DMV work.
I've taken in my smog certificate and paid the registration there and was given my new tags. Not sure about your situation though.
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