This is very easy to do with Endnote (select references in Endnote, press Alt 2), but similar functionality appears to be lacking in Zotero. Is this so?
Correct, that doesn't exist. This may eventually happen, but it's also a bit of a weird function, since it's fairly common to have multiple Word docs open and you can easily end up pushing to the wrong one, which I think has been the main reason for the hesitation to build this.
Obviously Zotero does have the Word add-on itself, where you can easily search and cite items in Word.
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Obviously Zotero does have the Word add-on itself, where you can easily search and cite items in Word.