You certainly could.It's a low-risk action with the following responses in roughly descending order of probability (assuming you're looking for a tenure-track job, and not just an adjunct/visiting position):
1) The e-mail is ignored. Chairs get many strange e-mails about things that are unanswerable. This goes into the e-mail folder labelled "To respond to when I have much more time than I do now"
2) The Chair writes back with a polite letter thanking you for your interest but that there's nothing in the immediate future, promises to write to you if there is, and then promptly forgets about your e-mail which gets archived along with the other people who wrote the same.
3) The Chair writes back saying that there's an opening for exactly someone with your specific research interests, teaching background, and desired seniority. They were going to have an open search and advertise this on all of the usual sites, but because you're such a perfect fit, they're skipping straight to hiring you.
I should add that #1 and #2 combined represent about 99.99% probability and #3 only happens once in a millennium. But again, the risk is so low that you could do it and buy a PowerBall lottery ticket at the same time and see what happens.
On a more serious note, many job sites have 'alert' functions that will let you know when a job that meets your criteria pops up. I'd subscribe to those. In the meanwhile, I'd become chummy with that department if possible so that you can hear about any gossip about spots. Or try to impress them so much that they decide to do a targeted hire with you in the spotlight.