If you do not have the required stickers to complete a quest, there are a number of ways you can collect them:
Random stickers are given each time you open a chest. 
A small number of individual stickers are available for purchase in the 'Shop'. The selection changes every 6 hours. Every day, only a small number of each sticker is available, so keep an eye on the 'Shop' to see when the sticker you need is available!
Achievement stickers can only be collected by completing their specific tasks, so make sure you check each game daily for those stickers!