The title is pretty self-explanatory. There is a professor who is an expert in exactly what I plan to study abroad, and I’ve taken several classes from him. He’s on sabbatical right now, but still in town and accessible.
Is it permitted to ask for a letter of recommendation while they’re on sabbatical, or is that one of the duties they get to shirk off for the semester they’re gone?
Yes, absolutely. Sabbatical is a release from teaching duties, not a vacation. He / she may be out of town, though, so if you don’t get a timely response to an email ask the department for better contact information.
Another academic dittoing what Dr. Drake said. Get in touch with that professor right away, ask politely if he would mind writing the recommendation, apologize for form’s sake for bothering him while he’s on sabbatical, and thank him prettily when he agrees (which he ought to).
Then lean on him politely and make sure he doesn’t forget the deadline.
[going off to write some overdue recommendation letters for a student of my own now…ouch! ouch! bad Kimstu! Lazy disorganized Kimstu!]
Just write it yourself, and ask him to sign it. The trick is to vary the compliments and writing style enough in three letters so that it looked like three different people wrote them.
d&r 
Actually, although the above was clearly meant as a joke, the most successful recommendation letters are based on something not too far off that. Professors really appreciate getting detailed information about what strengths the employer/foundation/internship/whatever is looking for, as well as gentle reminders about what the student achieved that’s relevant to those requirements.
So if your recommendation request includes something like “This program is looking for applicants with fluency in Portuguese and experience in biodegradable yurt building. Since I got good grades in three of your Advanced Portuguese classes and you are familiar with my work in the University’s Yurt Building Club, a recommendation from you would be extremely valuable in helping the program assess my application”, you won’t be exactly writing the recommendation letter yourself, but you will be providing some useful direction that I bet the prof will be only too happy to take advantage of.