There are other reports of something like Long-COVID from the vaccine, in a very small percentage of people. If COVID is triggering some weird immune response that causes Long-COVID, then the vaccine could trigger it, too. What we see with COVID is that the sicker you get, the more likely to get Long-COVID, but that people have also shown up with Long-COVID without having been particularly sick. To your immune system, getting the vaccine is very similar to getting COVID without getting particularly sick. Note that lots of people get noticeable immune response from the vaccine, with things like achiness, fever, chills, etc., which is essentially getting a little sick -- as most of the symptoms of viral infection will be from a combination of your infected cells dying from being overwhelmed by the viral replication and the immune response to the infection (some of the latter being exactly the same immune response that the vaccine triggers).
But note that (assuming Long-COVID from the vaccine is even a thing) if you were susceptible to Long-COVID from the vaccine, it's likely you would have been susceptible to Long-COVID from the actual virus. Only if you get the actual virus, you also have all the risks associated with an actual COVID infection, including a risk of even worse Long-COVID than you might have gotten from the vaccine. And the likelihood that everyone will regularly be exposed to COVID, considering how endemic it is, is essentially 100%.
So by all means, people might avoid the vaccine because they are afraid of Long-COVID from the vaccine. They should just be aware that (1) they will be exposed to the live virus in the wild, (2) that exposure is probably just as likely to cause Long-COVID in them as the vaccine and (3) they will also have all the risks associated with being infected with the actual virus -- up to and including death, but also just the risk of worse Long-COVID than might have been triggered by the vaccine. Their call.