When during the application process should a candidate mention that their spouse is also looking for a job
I'd mention it during an on-campus interview.

In my department, we don't factor "the two-body problem" into an initial evaluation (i.e., do we bring someone for an on-campus interview). So it's up to you about mentioning it in a cover letter or not. I don't think I have ever read that in a cover letter in chemistry, though.

During an on-campus interview, sometimes it comes up, and sometimes it does not. Keep in mind that usually, the department cannot ask. We prefer if it comes up, so that we can work on solving the problem at the same time we make an offer.

When we decide about who gets an offer, the two-body problem doesn't factor into the decision at all.

Why? Well, for one, we want the best person we can get. For another, almost all of our candidates have a two-body problem of some sort. Sometimes, it's another academic job. In other cases, the spouse is a doctor, or a lawyer, or another scientist or ...

Suffice to say, once we make an offer, we're trying to do the best to get the candidate to come, and that includes making sure a significant other would be happy here.

I'll be honest. We're much happier if we know there's a two-body problem (of any type) before we make the offer (i.e. during the on-campus interview), so that we can start working on finding another position. As I said, we want you to come. Sometimes, we don't find out until after we extend the offer, but that just makes our job more difficult because we have much less time.