Snapchat has embedded TurboVote promo videos between the Stories and Discover pages of the app, which you can access by swiping left on the main Snapchat screen after opening the app. The videos feature well-known celebs like Jimmy Fallon and Dwayne "The Rock" Johnson asking you to swipe up on the screen. Once you do so, you're redirected to a TurboVote sign-up page where you can begin the process of registering to vote.
TurboVote is a non-profit run by Democracy Works, whose policies state that no personal information you enter is sold or released in any way; it's solely being used to get you registered as a voter. So you shouldn't have to worry about getting any weird calls, emails, or snaps from advertisers.
The TurboVote sign-up feature is active until Oct. 7. If you don't see the promo videos in your Snapchat app yet, don't worry: They seem to be intermittent, so just check back later.