Hoo Ha's Jamboree is mostly a parody of The Rock-afire Explosion, an animatronic band that played in Showbiz Pizza (the competitor and later owner of Chuck E. Cheese's) from 1980-1992. The big beaver is based off of Billy Bob Brockali, the beaver cheerleader (and sole female character) is based off of Mitzi Mozzerella, and the rats in barrels are loosely based off of Looney Bird. The spider in the background may also be based on Antioch, a minor character from the Rock-afire Explosion. Hoo Ha Owl, however, seems to be based on Jasper T. Jowls from Munch's Make-Believe Band (Chuck E. Cheese's version of the Rock-afire Explosion).
There is a common misconception that they are a spoof of the indie point-and-click horror survival video game Five Nights at Freddy's. However, "Soos and the Real Girl" was written and in development long before the release of said game. Several crew members have also outright denied the connection, mentioning Rock-afire Explosion as the inspiration. In fact, Chuck E. Cheese parodies are a well-known trope. Any other connections are pure coincidence.
Additionally, the name was likely inspired by the Country Bear Jamboree, a famous Walt Disney World attraction featuring an animatronic band.