The COVID vaccine is not the first vaccine to be developed rapidly.
When Ebola struck West Africa, a vaccine was developed very quickly, not as quickly as COVID, but enough for it to have been used to control the epidemic at the time, and later recurrences of the disease in DRC and surrounding countries. I have not heard of any long-term effects. But, sadly, too many do not think more widely than the US and Europe when thinking about previous vaccines.
With COVID. although it was unknown to us until late 2019, it is part of a much wider class of viruses, known as Corona viruses about which a lot is known, both flu and the common cold are corona viruses and there was a huge amount of research into corona viruses in general and several in particular that could be drawn on, to give the search for a COVID vaccine a head start.
I think, all these claims and fears, that can easily be shown to be unfounded and allayed, if the person with them looks at a reputabe and reliable source like the NHS site, are really just a camoflage for deeper and more visceral fears about the speed of development in so many fields of endeavour that are hard to understand and the deep distrust of authority, whether, government, medical, technological, that alway grips peoples when they do not understand them. The 'They are all out to get us' feeling that often accompanies ignorance