Which is the proper spelling: "Adapter" or "adaptor"?
Solution 1:
Both are commonly used and listed in dictionaries, but adapter (with an e) is usually listed as the more common - see, for example, Merriam Webster or the Cambridge dictionary. It's not a strong basis for choosing one over the other, but it might be enough to settle the matter in your case :).
Solution 2:
I would tend to suggest in a situation like this where both spellings are considered acceptable to use the spelling that is most broadly accepted in the particular field you are working in. In this case, the leading example of usage of the word "adapter" is the description of the kind of system you are writing in Design Patterns: Elements of Reusable Object-Oriented Software by Gamma, Helms, Johnson and Vlissides, commonly referred to as "Gang of Four" or "GoF". This book (which is among the most widely-read books on software design there are) uses the -er spelling, so I would suggest that anybody else working on software adapters should also use that spelling.
Solution 3:
The British newspaper The Guardian, for instance, makes a distinction between adaptor vs. adapter in its internal style guide, namely, use adaptor when referring to devices and adapter when referring to people.