Not quite yet. The work has just started [1]. There's anecdotal evidence [2] that some people report improvements in long COVID symptoms after taking the vaccine (I have 0 background in chemistry/biology so I'll defer to domain experts to determine whether there's even a plausible mechanism of action for that).

I thought there were preliminary studies showing the vaccine is effective at prevention of infection, not just in reduction of severity of symptoms. Assuming that, a good prior to hold IMO (until we have such research) would be that it also prevents long COVID. However, since I can't find where I heard this, may be smart to withhold judgement one way or the other. That being said, what the sibling commentor pointed out seems plausible also to me to put into the prior (i.e. if long COVID is correlated with serious illness, then plausibly the vaccine would also prevent long COVID by reducing the serious response in the first place)

[1] https://news.yale.edu/2021/05/10/study-determine-how-covid-1...

[2] https://www.yalemedicine.org/news/vaccines-long-covid

Besides nearly eliminating the risk of hospitalization and death, the vaccines also greatly reduce the odds of contracting COVID-19 in the first place which logically would lead to a reduction in long covid. (contracting covid is a prerequisite for experiencing long-covid)