The studies we have on the vaccines are mostly all by the manufacturers (who stand to gain billions but will be sheltered if it goes bad) in which most of them are still in Phase 1 or barely into Phase 2.
In the case of these vaccines manufacturers were allowed to run multiple trial phases concurrently to speed up development. Phase III trials are really about better understanding clinical efficacy and when to use a drug in practice once it is Phase IV (which is just post market surveillance).
Also it is important to remember that most trial drugs don’t make it out of phase 0 or I, drugs often don’t show the promise that we hope for and we stop their trials early. Not making it to phase III or IV doesn’t mean that category of drug isn’t possible, it means that a specific formulation isn’t efficacious but that doesn’t necessarily make it dangerous.
It is not uncommon to have drugs given outside of trial anyway. We give compassionate exception drugs frequently even though they are still in phase II or III.