There is and has never been a US vaccine export ban, and the US has not seized vaccines meant for other countries. If I purchase an item before others, purchase by far the most quantities of that item, pay by far the most overall, and pay a considerable amount of money toward funding its development, I should expect that item before others. Any Kickstarter backer knows this; it's what the US and UK did, and what Canada and the EU did not.
The Trump administration last year signed gigantic contracts for every planned vaccine, because no one knew which ones would work. Like, enough for every American from one manufacturer, let alone the current four major available ones. More importantly, the contracts guaranteed the US the earliest deliveries.
By early June 2020 (<https://web.archive.org/web/20200603171013/https://www.nytim...>) the Trump administration had already identified and was planning to sign the aforementioned huge contracts with Moderna, AstraZeneca, J&J, Merck, and Pfizer. (An 80% success rate is fantastic in drug discovery.) By that time the US had already paid $2.2 billion to three of the companies. (Also note the skepticism throughout the article that any vaccines could be delivered anywhere within the timeframe the administration was promising.)
The UK signed a similar contract for the AstraZeneca vaccine. The EU and Canada did not assure themselves of such quantities. Canada also bet on CanSino because it was afraid that the US would ban vaccine exports (which, again, never happened). Of course, the Chinese did not live up to the contract.
Before you say "But what about—", the Trump executive order from December 2020 merely sets up the legal framework to prohibit exports if desired. But that does not mean that the framework is invoked. Let me repeat: The US signed contracts that were a) huge in size/scope and b) from every pharmaceutical company working on a vaccine, which c) got the country the largest and among the first deliveries. The UK did the same thing with the AstraZeneca vaccine, and spent a lot of money to retool domestic plants to produce it in addition to the non-UK doses it bought.
By contrast, look at Canada as counterexample. Consider Maclean's desperate attempt to spin its procurement difficulties (<https://www.macleans.ca/news/canada/where-did-canadas-vaccin...>). If you look past the predictable false eliding of the US's firstest-with-the-mostest contracts as an "export ban", the best the magazine can do is admit that
* all Canadian contracts with vaccine providers that actually delivered were signed between late July and late September
* all contracts were signed after the collapse of the CanSino deal, which Canada had loudly bragged about as proof of its savviness at obtaining vaccines ASAP and circumvent any perfidious US vaccine export ban (which, again, never happened), and which it has done its best since the collapse to pretend that said contract never existed
* the other contracts that the magazine cites as proof that CanSino wasn't the only basket Ottawa was putting all its eggs in are with VBI (Who?) and USask. They may or may not yet deliver effective vaccines, but it's all now a bit beside the point, eh?
(I don't disagree that to country B waiting on doses, the outcome is the same whether or not the cause is country A implementing an export ban or country A having bought up all the doses by having signed the contract first. But there is a difference.)
If the situation were different, might the US have implemented a ban on exporting doses already designated for another country, similar to what the EU did implement toward Australia? Perhaps. But, fortunately, the US never faced this issue, because of the huge amounts of money it invested a year ago and the contracts it signed with said money.