Yes, I am in Eugene (braving the terrible 85 degree heat), but I don't know any nearby dojos called the Budokai. I know of one place here that I think teaches Aikido, Judo, TKD, and JuiJitsu, but they (I believe) call themselves Best Martial Arts. They are associated with the California Aikido Association and are fairly nice people.
Calling any set of martial arts MMA is technically fine, but (as Ron says) it usually implies a combat oriented system. It would be a bit odd to see Aikido featured as part of MMA (though not impossible www.aikidog.com