While less specific than the term mentioned by @DavidSiegel (i.e. an "Antinomy"), "impossibility", "impossibility of performance", and "impracticability" are more frequently used. If the obligation is imposed by contract rather than by a statute or regulation, the term "frustration of purpose" is also frequently used. Even more generally, a situation like this is called a "dilemma."
More obscurely, sometimes this gets litigated in the frame of whether non-compliance with the law was a voluntary act, i.e. the actus reus necessary to establish a prima facie case of a violation of a law imposing a criminal or quasi-criminal penalty. The "choice of evils" doctrine could also apply although this is a much narrower doctrine excusing liability than many people believe it to be.
There are also a couple of other possible outcomes.
Sometimes a court would treat this situations as an implied ambiguity or inconsistency in the law that calls for the court to interpret the true meaning of the statute or regulation. For example, in this circumstance, it is implausible that the statute or regulation was intended to put good faith vaccine distributors in a damned if you do, damned if you don't situation where they would be fined no matter what they did because conforming to the law was impossible. So a court might conclude that the requirement that "If you don't use up your vaccine doses, you're getting a huge fine" contains an implied condition that this only applies if there are sufficient people who eligible to obtain the vaccination to use all of your vaccine doses (which as as matter of practicality is very likely to be true, so it is something of a false conflict as applied in most cases).
But it isn't at all uncommon for people to put themselves in situations where their on ill advised or wrongful conduct puts them in a position in which now, after their prior bad acts, anything they do will be a crime or a civil offense, and they are indeed damned if they do and damned if they don't.
For example, if you start to attempt to steal a car and see a baby in the back, you can either continue, and face kidnapping charges and car theft charges, or you can stop after you have already done enough to make you guilty of attempted car theft even if you abandon the crime at that point to prevent yourself for kidnapping the child. No matter what you do, you are stuck committing one crime or another. But you actions will determine which crime you will be guilty of committing. The fact that at the final moment of decision that determines which crime you will be guilty of, any choice you make will expose you to criminal liability, doesn't make these laws invalid, because this Hobson's choice (using the term only loosely) is one that you brought that on yourself.
