Is there usually a person at the undergrad university that handles it? Or does my recommender have to fill it out for every single program? Or something else?
There are a lot of possibilities. If there is staff available, it may just get passed to them. But that is probably less available than in the past (US perspective).
But, given that the professor saves a copy of one, it isn't much work to prepare another unless it needs to be tailored. Usually, though, the body of the message won't change much for a given candidate.
Usually, if you need multiple recommendation letters from the same person, they will submit more or less the same letter to each institution, with maybe a few modifications depending on whether the institution has requests for particular information from the recommender.
Most, if not all, institutions in the US will have an online portal where you'll put in your recommender's information, and they will upload their letter and whatever other information the institution requests (some will have something to the effect of "rate the student in these key areas on whether this student is in the top 1%, 5%, 25%" etc.) If you are applying to multiple colleges/programs within the same institution, they may only need the recommender to fill out that information once, but this is again institution-dependent.