I try to refer to the official documentation for supported APIs in the MS Edge extensions.
I did not get any officially documented API from the Edge extension development side that can query the latest version of the extension from the store.
You can also refer to the referenced link to verify the same.
I am not sure whether there is an API available from the Microsoft Store side to query the latest version of the extension.
If you think there should be an API for this specific purpose then I suggest you go to the bottom of this page and click the
This Product button under Feedback options and try to provide your feedback.
Thanks for your understanding.
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