I had one of my games linked on Twitter with a summary card and I noticed that the image it uses is a still of my game's cover, which is a gif, blown up a to larger size. Naturally this looks pretty bad because the gif is small and compressed. Would it be possible to have Twitter summary cards grab a screenshot instead of the cover when a game's cover is a gif, or is that stuff all generated on Twitter's end? Here's the tweet, for reference: https://twitter.com/Pizzamakesgames/status/930907738800054272.

No, Twitter cards get what they need from special tags in the header of each web page. It's just not always easy to decide what to put there. For Itch.io, covers were just an obvious choice, I suspect.