Welcome to Android Central! I moved this to the Snapchat forum for more specific traffic.
I don't use that app, but if you can change the tone, it would be in the system settings. Go to Settings>Apps, select Snapchat, then Notifications. Tap on one of the categories (and tap on the word, not the toggle switch), then see if there's an option to choose a tone. But keep in mind that some apps don't let the user customize the tone.