A vaccine isn't going to alter your immune system response anymore than digging in the garden might. Honestly, digging in the garden or eating some fresh fruit/vegetables/cheese will probably expose you to more foreign genetic material, virii and microbes than a vaccine ever would, thereby altering your immune response to a much greater degree. It's ignorant made up reasoning loosely based on ignorant made up useless meandering out dated philosophies.
The immune system is a learned response system. It's trained by exposure. That's the way it works. The specific way it is introduced to foreign material doesn't matter, except that certain ways are more likely to allow an infection to take hold.
You know what can really alter and possibly mutilate a body? A serious infection.
Edit to add:
Going to the gym or just exercising alters the body by making it stronger, adding muscle, but have never heard a religious person say that is taboo because that would be ridiculous. Vaccines make your immune system stronger in much the same fashion, through use and training. There's absolutely no good argument against them.
Would be willing to bet that: