Edit 2: Fixed it! When you send a friend request and he/she accepts it, they will instantly appear in your in-game friend list. However, if he/she just accepts it without logging on to GT5, such as if they are playing a different game or is in the XMB menu and then turns off the system, you won't be able to select any of his/her drivers through the remote race feature on the GT.com website until they log on to GT5 (you can still select the drivers manually through the game). As soon as they log on to GT5, their PS3 updates the server on Polyphony's end that the two of you are friends, giving you access to select each others' Bobs.
I'm assuming the side-bar on the GT.com website with a list of your friends is updated via the Playstation Network and the 'Remote Race' functionality is updated via the Polyphony servers, which would explain why they appear in your friend list on the GT.com website immediately but you can't access their Bobs for remote racing until they log on to GT5.