My lawn has been taken over by this viney weed. Does anyone have any advice as to a weed and feed that will work to kill it?
I am guessing that you've got more than one weed involved; possibly Spotted Spurge, Black Medic and/or purslane. All broadleaf weeds, very common. Easily selected for and killed by a 'Broadleaf Herbicide' for lawns. If you use a Trimec formulation in your 'Weed 'n Feed' make sure that you wet your lawn slightly before application. This ensures the herbicide sticks to the broadleaf weed leaves. I'd prefer using a sprayer to select specific areas to kill weeds.
Killing weeds is not going to solve your problem. We need to know what you are doing to maintain your lawn. Such as; how long do you leave your grass after mowing? This is a big deal. Too short, the grass wimps out and weeds will invade and take over. 3" is the MINIMUM height of this kind of grass. Vigorous grass can outcompete weeds and shade the soil so weed seeds have a hard time germinating. Watering deeply and allowing the soil to dry in between is crucial.
What fertilizer are you using, how often? Aeration, pulling plugs of soil out of the lawn and left to decompose...are you doing this once per year? Lots of other questions with great answers on this site you should check out. Let us know and send more pictures to show the amount of weed in your lawn. If this is the only spot, please use spot spray. Let us know what your maintenance schedule is...thanks!!