Shakespeare's Globe Archive is limited in scope to materials from productions at the reconstructed Globe Theatre in London from 1997-2016 and the Sam Wanamaker Playhouse from 2014-2016. However, it does include promotional materials. Some search tips for targeting those:
For some newspapers included in this Proquest News and Newspapers, the database includes page images that contain advertisements (usually not for the most recent years). Search tips for targeting ads:
These two databases use the same search interface. Tips for using them to find advertisements of plays:
If you have a particular production in mind and you know the name of the theater where it was staged, try a Google search on the theater. If the production was relatively recent, the theater website might still have some of their promotional information up. Theater websites may also have links to their social media accounts like YouTube (some theaters produce trailers for plays), Instagram, Facebook, and Twitter. Social media channels are another channel for advertising productions.
A Google images search is another possible way of finding posters or promotional images from a play. Again, try searching on the playwright and title, or the title and director (or theater) if you have a particular production in mind.