Will spraying morning glory with vinnegar/lemon juice/soap actually kill it or just burn off the leaves? Will it eventually die? How worried should I be about the morning glory in my grass?

I really doubt that vinegar will kill the roots. It will be interesting to see. I have some I was going to try the vinegar on. Morning glory is a tough one because of the horizontal root system that is about 6 inches deep. If you can get those roots on top to dry out they will die. Morning glory in a lawn that gets mowed often doesn't really do very well, but unfortunately the sod doesn't crowd it out either in most cases. Dig it up is about the only way to get it unless you want to resort to herbicides.