Our by-laws state that our Vice President will fulfill the duties until someone is elected... works great for us as the Vice President is already authorized to sign checks-- I would check your by-laws and do some re-writing to take care of this same question for the future... good luck!
I would definitely elect someone who's willing to do the job temporarily. Officially that person would be elected to the full term, but could always resign later. I'd also see what your bylaws say about filling vacancies; and if you have a vice president, see whether that person is willing to resign from that post to fill the vacancy at treasurer. (Organizations generally can function without a vice president, simply running the risk of losing the president while the vice presidency is vacant. Without a treasurer, you may not have legal authority to write any checks.)
Our PTO nominating committee was unable to find someone willing to be treasurer next year. We vote on next year's board this week. Our by-laws say nothing about who is responsible for the position's tasks if there is no one in the role. How would we determine this? Could we elect someone who would be willing to fill the role temporarily, assuming we might find someone in the fall? Any other thoughts?