At its most basic, you could use an IF test like:
{IF{MERGEFIELD Zip}<> 0 {MERGEFIELD Zip}}
or:
{IF«Zip»<> 0 «Zip»}
The above should work fine for 5-digit Zip Code data. However, if your source contains any 5+4-digit Zip Code data, you'll need the more comprehensive solution found under
Mailmerge US Zip Code Formatting in the
Mailmerge Tips & Tricks 'Sticky' thread at the top of this forum:
https://www.msofficeforums.com/mail-...ps-tricks.html
Note: The field brace pairs (i.e. '{ }') for the above example are all created in the document itself, via Ctrl-F9 (Cmd-F9 on a Mac); you can't simply type them or copy & paste them from this message. Nor is it practical to add them via any of the standard Word dialogues. Likewise, you can't type or copy & paste the chevrons (i.e. '« »') - they're part of the actual mergefields, which you can insert from the 'Insert Merge Field' dropdown. The spaces represented in the field construction are all required.
Once you've done the field coding, simply re-run the mailmerge.