The field is mathematics (including applied and computational mathematics).
While there are plenty of visiting assistant professor position on sites like MathJobs.org, I rarely see visiting associate professor positions, especially in USA. Is it because departments do not advertise these positions? Or that they just don't have such positions? In either case, how would one look for such a position?
At many research universities, "visiting assistant professor" actually means "postdoc with teaching duties". In other words, they are postdoc positions paid for by the department and financed at least in part through the revenues that come in through teaching.
The positions are not called "postdoc" because it is a "faculty position" so that the holder of the position is entitled to teach certain courses that only "faculty" can teach, and so that certain visa rules apply for candidates who come from abroad. The requirement for it to be a "faculty" position then prompted universities to call them "X professor". But because these are not tenure-track positions, one needs a different prefix in front of "professor" from the usual "assistant" entry-level position, and so universities chose "visiting assistant" to make it clear that the position is temporary. "Visiting" in this context is not meant to imply that that person has a position somewhere else and is only temporarily "visiting while traveling".
Given all of this, it is clear (hopefully) why there are no "visiting associate professor" positions.
My impression, which may certainly not describe all cases, is that "Visiting Assistant Professor" translates to "non-tenure track assistant professor" and fills a role in the academic hierarchy similar to what is called a "post doc" in some fields. Therefore, it makes little sense to refer to a "visiting associate professor", since "associate professor" typically means "tenured".
Someone could still have a "visiting"/temporary role while holding a tenured spot someplace else, but this isn't likely to be a job you advertise for candidates at large, it's a position created for a specific person, likely aligned with some source of funding.
VAPs seem to be more common in areas like math where post-PhD academics are mostly funded through teaching appointment (perhaps also in social sciences and the humanities? I can't speak much to this). In fields like biomedical science where most post-PhD academics are funded by the grants of more senior professors, these "VAP" positions hardly exist; I have only seen them in clinical settings where someone is given a "VAP" title to be a research post doc + teach med students in the clinic, or perhaps at (especially smaller) institutions who are trying to lure people in to be adjuncts.
Even before the pandemic, not only in upper-tier "research" universities, but also smaller colleges, in the U.S., in math, "visiting assistant professor" positions were/are mostly for very junior people who somehow did not have any better job offer in the current cycle, but/and will come and teach. Prompted by @Alexander Woo to be more precise: in some cases these positions do not have such an onerous teaching load but that maybe they can enhance their CV in a year or two, and then go back on the job market. In others, the teaching load is "normal" or worse. In fact, this ambiguity is surely worth tracking for people contemplating taking such a job: they're not all the same. Get details.
More senior people, who do not need any sort of temporary job to keep a paycheck while looking for something better, would usually just be "visitor", or not even have any title. This might include people, with tenure elsewhere, on sabbatical from their home institution, who may or may not teach at the host institution. If they are given a title, it might well be "visiting assoc prof" or "visiting prof"... but/and, often, by the time people get to such positions, they are full professors at their home institutions, so they'd just be "visiting professor". Hence, few visible "visiting assoc profs".
I think (a guess, mostly though I know of at least one case ) that those are mostly done by invitation. A professor wants to work with someone for a year and the person has a sabbatical year coming up, so they find a bit of money, etc.
Or the visitor seeks out a potential colleague and proposes a year of collaboration. If a sabbatical is rich enough the host institution may not need to provide much.
For non-permanent positions it may be that different rules apply, not requiring a search.
