When you download and unzip the Gerbers you will find a file with either a .DRL or a .txt suffix.
This is the drill file. It should be a fairly simple task to extract a drill table from this file.
To find out about downloading your Gerbers, please see:
http://easyeda.com/order
I don't know of a way to do this directly but you I think you could do it using a script that took the information from the .DRL file and merged it with a copy of the documentation layer.
This may help:
https://easyeda.com/Doc/Open-File-Format/
and:
**Super Menu > Miscellaneous > Scripts...**
**Super Menu > Miscellaneous > Run Script...**
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