I'd be surprised if any of the old cartoon shorts didn't have posters.
In the old days, animated shorts were shown alongside live-action ones and newsreels. It all added up to the movie-going experience. They were a part of pop culture then and they still are now, although in a different form. Remember, animation was time- and labour-intensive. So it was common to find animators/studios churning out any number of shorts, before getting ambitious and venturing into a feature-length movie. And even then they would continue making shorts. Why? Because these were relatively easier to produce.
These shorts were advertised to the public in pretty much the same manner as full-length, non-animated films were. Hand-distributed flyers, smaller posters in prominent locations, print advertisements and inserts in dailies and magazines (which came later) were all part of the usual advertising 'blitz'. But there always was advertising present at the venue itself, using theatrical posters and lobby cards.
In fact, Gertie the Dinosaur had vaudevillian and theatrical posters to promote the short. And that was in 1914.
Even if tight budgets restrained their advertising efforts, no studio would do away with theatre-based promotions. And there's a good reason for that. Theatres gave the studios a made-to-order captive audience, a group of people gathered in one location for a period of time. These people were familiar with your regular merchandise and they probably looked forward to your next offering. They were also in 'the right frame of mind' i.e. in the mood to watch a movie. To put it bluntly, these folks were susceptible to studio promotions. Think of it as in-store advertising, like the placards and displays in supermarkets.
Despite the rising cost of print and production, theatrical posters are still the most affordable and effective way to cut across to movie-going audiences. They're the staple of any movie's marketing plan. Look around you. From standalone theatres to multiplexes, the humble movie poster still holds its own against the most aggressive social media tools. All because it's in the right place.
So I don't believe that studios opted out of posters for their animated shorts. They were far too valuable to forsake.
Here's my theory: They became collectibles. And it's primarily due to this that such posters have all but disappeared from public view.
Full-length features left a larger footprint, they were distributed and shown in other countries. In a lot of cases, the posters used abroad were variations of the original design. Either way, it's easier to track down movie posters, including those for animated features. Shorts, on the other hand, had a more 'domestic' history. Shorts from the US stayed in the US, and if at all they ventured abroad it was to US territories and other venues like, say, US military camps. Similarly, shorts from Britain stayed in Britain and the same goes for the rest of Europe or any other country which made them.
The lack of an overseas release also meant that there were lesser prints in existence, lesser number of posters too. Over the years, a lot of those posters would have been lost due to neglect or ignorance. And since something rare usually fetches a good price...
You've heard of people shelling out thousands of dollars for plates and cels of animated movies. My guess is that there's also a booming market for posters of old cartoon shorts. Take a look at some auction catalogues here and here, and you'll see what I mean.
There you have it. I think that the posters you're looking for are tucked away in the vaults of specialised auction houses and collectors. And they've probably been there long before copies were made for reference or research purposes. Then again, that's just my theory.