Never, you’ll destroy SEO traffic this way.
The dealer website will provide a listing ID especially if using one of the big providers like anything made by cox, dealeron, CDK, dealer inspire, etc compare that listing id using do only when search = 0 if not 0 then create a new listing. After this process you’ll send it through a flow that deletes any car from your DB that exists in yours but not theirs.
Bare in mind if you’re going for mass adoption most of these car websites and inventory management tools use excel sheets hosted on FTP servers, kinda a pain, but if you setup that flow there won’t be any customer you can’t help in the dealer space.
I’ve spent a lot of years in that niche both developing and marketing, if you need help send me a message.