The term 'communist' is actually pretty old, here is an example of it being used in a 1777 philosophical book. Karl Marx was developing his ideas in the early/mid 1800's, and the English Communist League was established in 1847. So, it's entirely possible that a 'Communist Party' could have gotten votes in the 1876 US presidential election.
However, the only American party that could have been labeled as such (that I could find) is the Socialist Labor Party, conveniently established in 1876, which itself was formed from an earlier Illinois party. Everything I've found on them seems to refer to the political parties as socialist and not communist. It's certainly possible that somebody called them communists, or that David Leip is simply using 'communist' to mean 'socialist'.
I couldn't find (in an admittedly short search) any records of the party getting any votes in 1876, or even who their nominee was if they participated. 32 votes is tiny enough to not have been meaningfully recorded if it did happen, and hard to fully disprove if the claim is just made up. If the website doesn't site any sources for its info then you have no reason to believe it, though the claim definitely is something that could have happened.