Vermouth showing up, saying something to Gin we couldn't hear, and then Gin calling off Chianti after that is a pretty good indication Vermouth is involved. Bourbon is the one who is doing all the footwork and case solving, so it's mostly his effort, not Vermouth's. I think it would be weird if every bad guy in DC was a master of disguise capable of latex perfection, so I think it is a good call on Gosho's part to have Vermouth help Bourbon with this. Okiya's disguise is pretty simple if he is Akai: Glasses and a wig or hair dye + straightening. Regarding two people disguising seeming implausible plotwise, Gosho has already pulled the festival of disguise card once before for the Vermouth arc climax. We had Vermouth as Araide, Conan as Haibara, Heiji as Shinichi as the invisible man, Vodka as a zombie, Yukiko as Medusa, and the case culprit disguising the primary case suspect as himself. Okiya and scar Akai are small potatoes compared to the 30 disguise pileup that was Volume 42. But the quote's context implies that the detective she was talking about is the one who set up the plan to trick Gin and the Black Org by faking Akai's death. Since Conan was part of the fake Akai's death plan, it would make sense that Conan was the one. Gosho wrote/drew it like that to trick the readers into thinking Okiya sent it Mouri the text first. This is also the reason why scar Akai was not shown using his phone earlier. Then next file there is the shocking twist that it wasn't Okiya who sent it.