No, Jonathan Taylor has not requested a trade. Per the Colts' coaching staff, Taylor himself "hasn't requested a trade" (from the Jonathan Taylor page).
That said, there's a ton of smoke around him. The Colts (2-5) just fired their GM, shipped out Quenton Nelson, and are in full fire-sale mode. Multiple teams — the Vikings, Seahawks, Broncos, Packers, and Giants — are circling as potential buyers ahead of the November 10 deadline. Taylor's wife even publicly asked "Do I need to start packing?" amid the roster dismantling.
So while Taylor hasn't asked out, the organization might move him anyway. Establish The Run noted it's a "massive fantasy story if the Colts do decide to move on" — . From a fantasy perspective, a trade to Minnesota or Seattle would actually boost his ceiling.