It's not reasonable to expect every individual to have the time, resources or expertise to personally run a full chemical analysis on every dose of vaccine or medicine that enters their body.

So... you don't. You don't verify it, because you can't. You can't verify all of science from first principles. You can't inspect every building you enter and verify that it meets code. You can't inspect every electronic device for NSA taps. You can't inspect all of your food, drinking water, or even all of the FOSS code you use. Society is too large, and the domains of knowledge and expertise that feed into it are too complex for any individual to fully comprehend.

You can't eliminate risk or trust from life. As far as vaccines go, you just have to trust that they work and are not likely to be poison and take the shot.