Amazon.com
Despite their age, there are still copies of new, unopened games available on Amazon.com, as well as used copies sold by vendors.
EBay
There are plenty of PS One games available on the bidding site eBay, where buyers purchase their items from individual vendors instead of from the store. The site has search distinctions for genre, quality, ESRB rating and price range.
JJ Games
This website specializes in selling previously used games and provides a decent selection for those on PlayStation One. The search can be filtered by quality, and also by whether the original box and instructions are included.
BuyPSGames.com
The aptly named BuyPSGames.com doesn't sell the games; instead, it acts as a go-between for Amazon.com. The advantage is it's much easier to search.
Shopping.com
If you're having trouble finding a specific title, you can search on Shopping.com. The website will find you websites where you can buy the game from, and also give you the compared prices so you know where to make the purchase.