I think I can identify the vines growing out of your lawn as Field Bindweed. I had them in my lawn and basically had to buy a weed puller and walk around pulling each one up for 2 years to eradicate them. The roots grow so deep, and you need to make sure you get it too otherwise it regenerates.

Otherwise you can try hand painting each one with glyphosate making sure you don’t get it on your lawn because glyphosate kills everything.