Gary Sheffield has 11 career walk-off hits. We're gonna factor out Willie Mays and Duke Snider because available data (from the likes of Baseball-Reference and MikeMav.com) begins in 1954. Among all those others who have played for the Mets, who has the most career walk-off hits, with 19?
The answer: Rusty Staub.
Here's a more thorough list with help from MikeMav

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The most walk-off hits by someone who played for the Mets (excluding the incomplete data on Mays and Snider):
Rusty Staub with 19