She claims to do so in an interview with Aya, but it's never made clear as to whether she was serious or made it up to threaten her. According to Akyuu's records she does claim to run a yakitori stand to others as well.
Rathurue said: Think this picture as a friendly visit from a rival stand-owner-scene.
That much is fine, but the actual problem is that Mystia despises yakitori. In fact that's why she got into selling eels in the first place. To Mystia Mokou would be a monster.'^'
lamusa said: wait wait, mokou she is work yakitori resturant right?
Chances are, that's something she made up in her interview with Aya to scare the Tengu away to keep her nose from prying into places where she doesn't belong. At least from her tone, it was as if she was trying to remember what Kaguya said earlier to bring up the yakitori. In any case, I think it's a rumor that has spread pretty quickly because Mokou is so reticent about herself, the yakitori is just about the only personal thing she gave away (and even that may have been a lie.)