Hi
Snapchat does not offer an app for Apple Watch, so cannot be installed on it:
If you would prefer that it did, then you may wish to submit a feature request to the app developer.
Notification mirroring settings for third-party apps from your iPhone are available on your iPhone, in the Watch app, under My Watch (tab) > Notifications:
See and respond to notifications
Also be sure to check your notification settings for Snapchat on your iPhone, enabling all desired options.
How notifications behave depends on the status of your devices and your settings: