I've seen this pop up at parties and on late-night YouTube binges: yes, there are karaoke tracks meant to match the 'cup song' lyrics (the version everyone knows from 'Pitch Perfect' is usually called 'Cups' or 'Cups (When I'm Gone)'). What trips people up is that the iconic thing about the cup version is not just the melody, it's the percussive cup rhythm that sits between the sung lines. Some karaoke instrumentals include a percussion track that mimics the cup, some don't, and others include a simpler clap or beat instead.
If you want something ready-made, search for "Cups (When I'm Gone) karaoke" on YouTube, Karafun, or platforms like Karaoke Version — you'll find instrumentals in a variety of keys and tempos. A lot of those uploads try to preserve the call-and-response phrasing so the lyrics line up with the cup pattern, but not all of them nail the exact rhythm. If you want an exact match, look for versions labeled with "cup rhythm" or "cup percussion"; community uploads and covers are often the closest to the original cup-game feel.
When I'm practicing for a small get-together, I usually pick an instrumental at the right tempo (around what feels natural for me) and then either mute the percussion channel if it's too busy or record the cup pattern on top so it syncs perfectly. If you're planning to perform, a little DIY — lining up a click track to the tempo and practicing the cup pattern with the backing track — makes everything sound way more cohesive. Honestly, hunting for the perfect backing track can be half the fun.