if you had quality.. predictable named sunflowers ... these seedlings may or may not be like their parents ... at that size .. you could use a hand trowel .. and spread them all over the yard.. its called transplanting ... but for the weather of course .... since you sound a bit like a newb ... i also find watermelons seedlings.. where the kids spit seeds off the patio.. lol ... pumpkin seedlings where they cleaned out the Ps ... etc ... gourds where they dropped decorative fall gourds .... and just about every weed known to man.. under a birdfeeder ... a lot of this stuff is near bulletproof ... which is one reason i call them weeds ... its usually us.. who kill them with too much love and care .. ken

Realistically, you have two choices with a well grown invasive plant like this bittersweet. One is to cut to the ground all vines and on the fresh cuts paint a brush killer. Yes, it uses herbicide, but in smaller, very targeted use. This might take one repetition, but I haven’t had times when even a well established invasive takes more than one repetition. Otherwise, cut all the stems, and as soon as it resprouts, remove them before they are fully leafed out, and then repeat for the next two or more growing seasons every week or so. The goal here is to prevent it ever regrowing enough leaves to create energy so that the plant uses up all its reserves and is unable to replace them. With bittersweet, new stems won’t just grow from the main stem area, but will sprout from roots as well. If you let it establish leaves, it will start replacing the energy it used up trying to regrow, so this will take persistence and consistency. When a plant is really large and inter-grown with other plants like this, you won’t be able to remove all the roots with anything less than a tractor or bulldozer.

Rubus of some description. A bramble. Prickles are just visible on the top one.

If you don't want to apply herbicide to the stump you could cover it in a piece if thick black plastic well weighted down to exclude light from any regrowth. Leave it in place until this time next year at least. Just out of interest the berries are edible and used in Iranian cooking. They're dried and taste quite sharp. Oh, and it's not a vine. It's a shrub. Your example is just a bit straggly.