When I do accessibility checks for emails in which I include emojis, those emojis are never flagged as being an issue.  Like, it doesn’t prompt me to include alt-text or anything.  So are emojis accessible?

They are! At least, when used reasonably and judiciously, they’re fine. There’s no need for alt text because they’re technically not considered images, but characters just like any letters of an alphabet (similar to how a kanji or Arabic character isn’t an image), and they all have specific names that are read by screen readers. You can look up the name of any particular emoji on Emojipedia to find out how it is read by a screen reader — for example, 🤣’s official name is “rolling on the floor laughing”, so that’s what the screen reader says when that emoji is used. This is also why I say “when used reasonably and judiciously”, though. If someone is in the habit of using lots of emojis as emphasis, every emoji used is read out. So if something is really funny and someone types “🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣”, that could get read out as “rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji rolling on the floor laughing emoji “…so I’d recommend not using long strings of emoji.