Unfortunately, there is no way to currently purchase a specific mascot. You can find a daily rotation of mascots in the shop each day, and you can refresh that rotation by spending a few cookies.
Other options to try to acquire a mascot you really want is to ask your league mates if they'd gift one to you. Maybe in return, you could gift them one of yours.
Over the course of a year, we may promote and offer free mascots on social media. But if there's a specific one you really want, please try to check the shop once each new day you are using Sleeper!