I'm playing someone in a tournament who makes many moves daily, but hasn't moved for me for many days. His vacation icon has also been up. How does that affect his timebank?

His timeclock keeps on functioning normally, and could even run out, but the skull will never appear for you. Vacation flag is ONLY protection against timeout, it doesn't stop or pause his clock. He will have to move before his flag comes down, or else risk a timeout loss.