“Sign in to download from the App Store” dialog does not allow paste
You can use Keyboard Maestro with a macro such as this which takes the current clipboard and inserts its contents by typing it rather than pasting, allowing the clipboard to be entered into fields where pasting is forbidden.

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Also, for what it's worth, OS X 10.10 Yosemite does let you paste into the aforementioned field and I believe so does 10.9 Mavericks.