Ivy a nuisance.
You will need to either dig up the entire system or use an herbicide on the plant. Use a broadleaf weed killer/herbicide on the vine. Make sure to spray the leaves of the plant. This herbicide will not kill the grass as it will kill all non grass plants. So be careful if using this in your gardens as it will kill the plants in your gardens if sprayed on them. You should read, understand and follow all the label instructions on the herbicide. You can protect nearby plants when applying the herbicide by covering them with plastic, a sheet, or a box and remove this as soon as the herbicide is dry on the offending vine. There are also some herbicides specifically designed for tougher plants/vines and these may also be painted on the offending plants, again read, understand and follow all label instructions. Thank you for asking the eXtension site your question.
Thank you for your response. Is this an ivy or a vine, like you seem to be saying? This only gets to be about 16 inches tall. It reminds me of "snow on the mountain."
It could very well be the non variagated snow on the mountain, after looking at the leaves it seems to be that. You have a battle on your hands. This plant is even more aggressive than the regular snow on the mountain. Use a broad leaf herbicide on the leaves, keep after it. Any roots in the ground will send up leaves/shoots, but if you keep killing them, eventually the energy in the roots will be depleted and the plant eradicated. Good luck.
Would it be okay to go ahead and plant stuff in the area they're growing and just dig them out of where I would be planting?
It would be ok to do that, as long as you understand that if you miss any of the roots they may sprout. So be very observant for the rest of the growing season and kill/pull any new ones you see coming up. If you are not planning on using an herbicide you can plant as soon as you are done removing the weedy plants. Good luck and happy planting!