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I mean, it's a hard design problem to solve. Assuming you don't want to force use of voice chat, text is the usual go to for multiplayer games. Some ideas I had were to slide open the chat right after a minigame so there's more time, maybe text from the chat scrolls onto the other people still in the minigame like NicoNico and it becomes a funny distraction. While that's happening(when you're done but others are not), you can see what the other players are doing in a big collage or some such. Besides that, you could leverage the mouse primary gameplay and instead of chat use freehand drawing to communicate and still do the previous things I mentioned.

Since the minigames are so short, one way to give more time for typing is to chain several minigames at once, so those who finish fastest have more downtime waiting for the slow guys.

Though, now that I think about it party game collections like Jackbox don't have chat I think, so it could work fine like that too. I'm sure one of these options will jump out as correct depending on what you envision the final experience to be.