It is very difficult to kill morning glory and you need to be vigilant over a long period to completely eradicate it. Just spraying something like Roundup probably won't work.
You obviously need to get rid of all the roots but it is not easy as they can spread over a large area and pulling them out is not advisable as even if you leave a small part behind new vines will shoot. So cut back all the plants then use something like a blackberry killer on the roots and keep spraying any new shoots that pop up. I have also read that Weed and Feed for Lawns (MCPA and Dicamba) kills it but can't confirm.
If you can cover as much of the infected area with black plastic, weed matting or carpet it can also help smother the plant. - greygardener
Be careful with how you dispose of cuttings, too. Don't leave them around, or put them in the compost. Best thing to do is to place them into a plastic bag and dispose of it in the rubbish. - Isobel
You will not be able to use the area as a vegetable garden if you use Roundup, blackberry/tree killer or Weed and Feed. You can try boiling water, which will kill the plant, worms and all beneficial microorganisms. But you can build up the area by adding in compost, organic matters without worrying about the poison. - QuailFlock
I was a farmer in a past life and attended many chemical handling courses but I am definitely not an expert, but think I may be able to help here. I use old jars to put my herbicide mix in so I don't affect surrounding plants or the soil but do this when not expecting rain so you don't get overflow. Usually Roundup or similar with a bit of brush or bush herbicide Dicamba or metsulfuron and some water.
Use a really strong mix (depends on what you are using and how big the plant is) and gather up a handful of the most vigorous recent growth of vines. Then twist or scrap the ropes of the vine gently so they are scratched but not broken and push that into the jar. The idea is to get the stems to absorb the mixture which it will circulate through the plant and eventually kill it but take care where you place the jar away from kids, dogs, etc. and perhaps elevated on a fence or tree with some wire. Works a treat and over a couple of years doing this once a year I have eradicated mine when it used to cover my trees like a canopy. - Cathstarr