Upon installation of a new radio or a new alarm system could door locks stop locking after the engine start on a Subaru Forester 2000?

Yes they could. They would have disconnected your battery when installing these new products. Some Subaru's came from the factory without the option enabled. When the dealer has the tech do PDI (pre-delivery inspection) he can set that. I was a salesman for 3 years and had two request that option to be turned off. I asked the guy that did the PDI's and he told me how to disable it and told me how to enable it also but it's been so long ago.