I have a game where my opponent hasn't played since Sep 3. Multiple timeouts have expired only to be “renewed” for another 24 hours.

It depends upon how much vacation time your opponent has.  Vacation time will kick in automatically.

Your opponent may have as many as 15 vacation days.  (Vacation time used to be 20 days but I think it is now 15 days.)

So after he/she initially timed out (10 days) you might have to wait an additional 15 days.

UNLESS your opponent is a paying member.   Paying members get 30 vacation days.

Note:  Richard's FAQ says non paying members have 15 vacation days, but when I click on my Vacation Status, it says non paying members have 20 vacation days… so I don't know which is correct.

Just recently, a relatively newly registered player used up 21 vacation days against me (yep, I know it says that you have 20 vacation days to begin with, but the counter goes down to -1, so you actually have 21). I don't know if the 15 days in the FAQ is a typo or if that is implemented for new players from now on, but regardless, you are probably looking at 21 days against someone who is not completely new to the site. That plus possible delays in the triggering process.

In OP's case though, the opponent in question did actually register now in September.  It seems that he has had 7 vacation days triggered so far (and is currently at +15 hours). So we could be looking at either around 9 more days, or around 15 more days before he loses on time, depending on the truth of the FAQ.

Sometimes, vacation time resets during the year. I have never heard what causes it.

Richard sometimes manually re-sets vacation time when there's been an outage. Other than that, it only re-sets at the beginning of the year.