How do countries obtain US dollars when adopting it as their official currency?

There are many different ways to acquire the US Dollar. When a country decides that the US Dollar be the official currency, depending on the nature of the country’s relationship with the United States, the US Government would most likely designate a commercial bank, most likely out of Federal Reserve New York jurisdiction to hold the US Dollar balances for the country. The country in question must have some tradeable assets that can be converted to the US Dollar (oil? gold? US treasury bonds? previous UD Dollar reserves? exports? other currency reserves like GBP, CHF, EUR, JPY, etc.)

If the country in question does not enjoy a direct US Dollar country-level correspondent account with the bank designated by the US Department of Treasury, then the situation is slightly different.

Let’s take a country like Venezuela, where, right now, Venezuela's official currency, the Bolívar (VEF) is essentially worthless. So the stable parallel currency preferred by everyone else is the US Dollar.

There are many ways the US Dollar can come into Venezuela. Legally if a Venezuelan bank that has either USD deposits or other currency reserves and is not sanctioned, etc. can buy physical US Dollars from companies such as Travelex or from banks like Cit or HSBC. Yes, physical dollars that are flown in. It is a very common practice.

Commercial banks can make a very educated assessment of how much physical currency notes they want to buy which will then hopefully circulate in the economy and they do just that, so as to jump-start the economy or become a catalyst for such dollar-based transactions. Once they see banks can provide access to currency notes, then bureau de change also start doing the same (asking the banks to buy notes on their behalf), informal dollars fly in via people who are visiting or bringing US dollars physically because they might not trust the banking channel or the government (read: taxation).

It is all about acceptance. In any of the scenarios, there are a handful number of players who work with central banks worldwide to provide access to physical currency notes.