Can you treat more than one tooth abscess at a time?

Sure you can: But it really depends on the details. Discuss this with your dentist.

Depends on cause: Abcessed teeth may require root canal treatment or extractions if they cannot be saved with root canal therapy. Multiple root canals and extractions can be performed if that is the recommended treatment by your dentist. If you can not get to your dentist right away, you should be on an antibiotic.

Certainly: In fact, it is best to treat all at once when possible. This helps get rid of the infection going into your entire system. Go for it !