lay a piece on the ground from the live plant and use a "U" pin (I cut them from hangers) to secure it in the soil and let it root. Can take a long time, maybe a season.
To get it rooting a bit faster you can try carefully knicking the back of the vine, adding the smallest touch of rooting hormone and then pinning it.
The key is not to cover the vine you pinned to the ground with soil, just let it sit there but it can't move around. Likes to be medium moist - not soggy, not bone dry.
We do this at our church garden and it takes about 3-4 months to get some good transplanting roots, from there it takes another 6 months to get a "plant".

GGG

I picked a bunch of wild bush honeysuckle vines and was planning on transplanting them to my garden. I had talked to a woman that had told me to put the vines in water and let them sit to root. Have you ever heard this? I've had the vines sitting in water for about a week now. Since picking them I've read a couple of places including above not to get the vines too wet. Now I'm wondering...

If it's the white and yellow honeysuckle vines, don't even bother! They are invasive and they will eventually eat your garden and frustrate you to no end. I know, they smell great...I just go for a walk and enjoy the lovely smell where they grow wild.

I snip some off, strip off lower leaves, dip in water, then rooting hormone and stick it in a small bed of peat moss/sand mixture that is located under the air conditioner drip pipe. This keeps it moist and it is ready to plant next spring. It has always rooted easily.