Hi;
Those do appear to be mole tunnels. There are all kinds of products on the market to get rid of moles, but I have found that the most reliable method is the old fashioned mole traps. But those need to be set very close to where you think the moles are tunneling or where you think they will tunnel next.  

Avoid any type of mole deterrent that contains a poison if you have pets-especially the gummy worm ones.

Finally, as with any wildlife issues, the most sustainable way to deal with them is to determine the reason why they are there in the first place and correct that and/or learn to accept an occasional visitor. Lots of moles may mean they have found a lot to eat there. Sometimes their target is grubs, but they also eat other insects and worms. But if you lift your grass up and find evidence of lots of grubs, work on improving the health of your lawn (such as keeping grass height at a height of 3-3.5 inches), and that will reduce the grubs. But moles eat many types of things. They do move to other locations in search of food and will likely not be around all of the time, so even a bit of patience may help with the issue too. The good part is that they are primarily carnivorous and will not disturb the grape plant roots or stems. In contrast, be on the watch for VOLES, which are different from moles, and WILL eat the bark off the stems of plants, including fruit plants.  Here is a link on voles so you know the difference:

https://extension.psu.edu/voles