When you have access to Splash-Cam you will get 7 animated stickers!
As you complete Missions and open specific fish in your Aquapedia, their stickers will also become available.
You can get more stickers and frames from releasing species. When you release them there is a chance that one of the chests they leave has a new sticker or frame for you! There are a ton to collect, so release as many species as you can for a chance to collect them all.
You can also use Splashcash to buy the stickers or frames by tapping on the shadow outline and then the Splashcash confirmation.
Some events have sticker pack rewards!