Photo's are taken by Cameron of your avatar and the Disney cast. They come up through fulfilling requests and throwing parties.
Earning a PhotoEdit
There are a couple ways to earn a photo opportunity with a character, and you don't have to take it when it does, but it will continue to show up until you do.
One way is to throw a Party in your Cafe themed after the series that the character is from. There are several cast members from each film that will show up from the main protagonists like Belle and the Beast to the side characters like Lumiere and Cogsworth.
It IS random, and no amount of resetting will change who is going to show up. At the party, there is an absolute chance to get a photo with whichever character(s) show up.
The other way is by fulfilling requests. After 1-3 requests, the character in question will have a little icon of Cameron in a speech bubble above them, signifying that they would like to take a photo with you.
Pete is the greatest example. There are others, but Pete is the one to fear. He shows up randomly and has no current way to summon him even though his Photo is one of the Sticker Achievements. Certain characters do not have a way to bring them in to your town and will show up randomly with a chance of having a request. They follow the same 1-3 rule though, thankfully.
Take a photo in your game by pressing L + R at any time. Sometimes, you’ll even get special photo ops that let you choose your outfit and your pose! Your photos will be saved to your system's SD card.