Is this a woody vine in the area in the forefront of the images? Is the same plant climbing the chain link fence? Or are these suckers from a nearby tree? I can't see it well enough to identify it. 

Is this area near water?  Or is it in dry shade?  

It's hard to make weed management recommendations without an ID and the growing environment of the plant.

I can tell you, though, that there are no herbicides that you can use that won't also kill your vinca if you don't take precautions and cover it.

One method that's suggested in situations like this is to mow the plant down repeatedly until the roots are exhausted and the plant fails to thrive.  This method works for stubborn weeds like horsetail, too, but it takes time and a lot of patience.  However, it won't work if these are suckers from a nearby tree.

Do you know what the plant is?  Can you send good closeup photos?