Ended up disputing the charge with my CC issuer. NFW am I going to jump through hoops on this.
Contacted the 3rd Party provider and they couldn't fix the problem in real time. They had me send screen shots of what I was seeing on their system. Also confirmed that the "refund" would be a store credit.
I don't know if all mailorder Toyota parts sites are doing this BS now, but I will absolutely be checking in the future. What if somebody placed a large dollar order and had a problem? They're not getting a refund... are you friggin kiddin me??? My problem was only $110, but I'm still going to fight to the bloody end for that amount. Can't imagine if this was a grand, or something.