The first thing that should be stated is we don't actually know what's in the vaccines. You can go and look in the vaccine inserts, look in the CDC's pinkbook of vaccine excipients, or research manufacturing methods to try to predict potential contaminants from using peanut oil suspensions, chicken eggs, aborted fetal parts, HeLa cells, monkey kidney cells, dog kidney cells, and so forth (and consider this in the context of allergy). But, you still don't really know. Hence the SV40 contaminated polio vaccines, the Cutter incident which just happened to occur as work on weaponizing polio was going on at the Rockefeller Institute for Medical Research. So forth.

Vaccines have three main pathways to yield downstream outcomes.
-Direct effects of the component meant to have antibodies produced targeting it.
-Direct effects of the adjuvant and excipients
-Effects via the immune system and overall reaction to the above.

There are various mechanisms.
-Microglial priming
-Immunoexcitotoxicity
-Immunogenic damage and other immune mediated functional changes
-Nonlinear dose response to excipients or adjuvants (such as aluminum at micro and nanoparticle scales)
-Synergistic action with other conditons and environmental pollutants, such as nanometals and radiofrequency radiation
-A cascade of system failures, forming self sustaining pathologic feedback loops. eg direct autonomic damage or disruption of proprioceptive fibers leading to pinched nerves, downstream a change in gut motility with eventual "leakiness". Experiments in mice show lesioning of the vagus nerve causes increased intestinal permeability in a matter of hours. This would cause a low grade chronic inflammation, autoimmunity, so forth.
-Contaminants, insertion of genetic material (eg "magneto protein" expressing genes), nanotechnology, biomimetics, synthetic biology