This is not true for Covid vaccine, or I think most more newly-developed vaccines.
In order to sensitise the body against a pathogen, we just need to dump into the body some of the proteins on the surface of the pathogen. If the immune system learns to recognise those proteins as foreign, they'll attack anything with those proteins on, thus the virus. In the old days, lacking our modern biotechnology, this was commonly attenuated or dead viruses. The proteins can be recognised, but the carrier is much less harmful.
With modern molecular biological techniques, it is relatively trivial to identify the proteins of a pathogen, and then genetically modify a safe carrier to express one or more of target proteins. Traditionally this vehicle would be a safe carrier virus (e.g. an adenovirus): the newer mRNA vaccines are very similar in concept to this, but more stripped-down (basically just an mRNA, or protein-printing instruction, in a simpler, lipid nanoparticle).
Although vaccines should be safe in the sense that they are not active, dangerous viruses, they should generate an immune response. This may cause someone to feel generally crappy in several ways, or cause some specific inflammation ("-itis") in places which can be a health concern. It is not necessarily true that if someone had an adverse reaction to the vaccine, they would have had an adverse reaction to infection with the virus. There are a lot of variables involved.
It's also not like homeopathy. Homeopathy is diluting a substance until there's effectively none of it left in the vehicle solution (presumably with the idea that the water sort of "remembers" the chemical). Vaccines actually need a substantial dose of stuff delivered.