Gosh darn it all to flipping heck. I figured out my main mistake from yesterday and tried again.. I think I'm really close to answering this question by using Stathead, but now we may have a slight problem with definitions, or possibly software bugs, or both
In several cases, Margot has come into the game as a pinch-hitter, but then stayed in as the defensive replacement. When I ask Stathead for a streak and specify pinch-hitting appearances, it seems that it only looks at games where the hitter came in and then was immediately replaced the next inning (most often for the pitcher, historically).
For anyone who's paid for Stathead, this link may give you my starting point and let you experiment.
For almost everyone else probably, you can have a look at the results here.
But to summarize: the player with the longest hitless streak when "strictly" pinch-hitting is Charlie Gilbert, a 1940s journeyman who flailed his way through 50 fruitless attempts. Or maybe it was 57 - there could be a bug in the database or the software that presents itself. Oh goody, another bug report to send to my good friends there. (It's a SABR project and I do like to support it every way I can.)
So maybe Charlie Gilbert, maybe not, but our good friend Margot is likely very far from setting any all-time records.
Second on the list is Mike Mordecai from 1997 to 2000, playing for Atlanta and Montreal, with 40 hitless appearances.
Our other good friend Billy Martin shows up high in this list too, at 33 failures spread across 1950 through 1961. You know, the more I look at this, the less I trust the results - would the Yankees have kept trying him so many times? I'll have to get back to y'all on this; it will likely take days and I know you're all waiting breathlessly. 😀