Trumpet vine readily suckers, so your best option would be simply to dig up a sucker with the roots intact. Otherwise, you'll probably better have results with air-layering than simply trying to root a cutting. By far the easiest way to propagate morning glories is from seeds. Be well aware that both of these vines are rambunctious growers. Morning glory is a pest in Florida for example.
I was thinking about trying the same thing, but read in too many places that Morning Glory will take over and become uncontrollable. Proceed with caution. I pulled them up and put them in the trash!