None of the vaccines contain any ferrous materials. Even the metal compounds in it are in the form of sulfides, none magnetic.
Even if the injection contained anything capable of being ferrous, or magnetic through an electrical system, the injection itself does not contain enough mass. A bulk of the injection is inert stabilizers and saline.
A magnetic solution strong enough to produce a localized effect that was noticeable would likely cause other problems.
As for any sort of electromagnetic systems, nothing I know of could be injected that would produce an electromagnetic effect. Those need power, after all.