Hi there jaycee297,
If you're not receiving Snapchat notifications on your Apple Watch, there are a few things you can check.
Is your iPhone unlocked or locked when this happens?
Notifications go to either your Apple Watch or iPhone, but not both.
If your iPhone is unlocked, you'll get notifications on your iPhone instead of your Apple Watch.
Learn more here: Notifications on your Apple Watch
Next, if you open there Apple Watch app on your iPhone, tap My Watch, then Notifications, are they turned on for Snapchat there? This link provides details: Change notification settings on Apple Watch
Thanks!