Honestly, the solution depends highly on your circumstances. Microsoft SQL has a concept of a distributed transaction, but I'm going to assume that "4 disparate legacy databases" aren't Microsoft SQL.
Here's the high-level approach I would take:
1: Understand which databases serve what, and what the needs for accuracy are. (For example, a database serving their public-facing website can probably be a few seconds behind compared to a database handling real money.)
2: Figure out which database is the source of truth. Update that database first.
3: Update the remaining three databases in a loop: Update the database, then check to make sure the update happened
You will probably need to add some state management where you put yourself into an "updating" state, so that if the update breaks in the middle, you can go back and recover.
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Another approach is to refactor the applications that are critical (Those that handle money, not the public facing web site) to always read from the database you decide is the source of truth.
Here's the high-level approach I would take:
1: Understand which databases serve what, and what the needs for accuracy are. (For example, a database serving their public-facing website can probably be a few seconds behind compared to a database handling real money.)
2: Figure out which database is the source of truth. Update that database first.
3: Update the remaining three databases in a loop: Update the database, then check to make sure the update happened
You will probably need to add some state management where you put yourself into an "updating" state, so that if the update breaks in the middle, you can go back and recover.
----
Another approach is to refactor the applications that are critical (Those that handle money, not the public facing web site) to always read from the database you decide is the source of truth.