UPDATE: Good news and bad news. Good news, I'm able to redeem points for gift cards again. Bad news, support didn't do anything, and I have no idea what resolved the issue.
I tried redeeming points for a gift card again, and this time I got a different error - not the one about the account being locked. I opened a new ticket and waited for a response. I didn't receive a response, but I tried again a few days later. This time the system asked me to provide a phone number to verify the account. After setting up the account with a phone number, I was able to redeem points for the gift card.
What actually fixed the issue? I have no idea. The best suggestion I have is to add a phone number to your account if you don't already have one, but this is no guarantee.
Good luck everyone.