Trump didn't invoke the DPA for vaccine production because by the time the vaccine was ready to be mass produced he had already lost the election and no longer gave a . Trump invoked the DPA plenty of times for ventilators, PPE, tests and other things. I don't see any reason why he wouldn't have done it for vaccines had he won the election, especially given how he would have been the one to get the opportunity to go claim credit (much of it undeservedly so) instead of Biden.
Trump's a and his fan club are idiots, but there's no way Trump was going to pass up an opportunity to do something easy that he could brag about.