"Trust the science, don't do research" isn't really a thing that I've seen from the supporters of infant vaccination. Typically, "trust the science" means that there's a broad scientific consensus on something. Doing individual research is fine, but knowing the limits of your own expertise is important. In the anti-vax community, I often see people stressing the importance of doing your own research or questioning authority, but then they latch on to poor-quality research, or misinterpreting results because they don't understand how to read them correctly (see: the entirety of the movement that believes vaccines cause autism).