The US government made it free to citizens. But nothing is actually free.

When some nation donates 1-million doses, there will be a market value for those doses: maybe $20 million for the batch or so.

It's not even black market vs grey market. This is just standard market economies between nations.

People will pay to skip the queue, which creates a perverse incentive to hoard and limit distribution to incentive paying.

Corrupt officials receiving free vaccine shipments could very well do this.

I assumed we wouldn't be able to supply 100% of the globe at the same instant, maybe we'd distribute ~20% in a first wave (to the most vulnerable people). That means that those who are rich[er] but don't meet the criteria are going to be in a position to want to buy.

If the country/county/area/suburb next door is poor, and a locality has lots of cases then I'm guessing anyone who can afford it has an incentive to acquire a dose of vaccine early, or a booster dose, by buying out someone in that neighbouring locality.

You need strong local government without corruption to distribute vaccines fairly.