Vines grow down from "root" plants in starbound, and I believe they regrow given time. Seeing as these are a good source of plant fiber for bandages, is it possible to move the "roots" so the vines can be farmed?
I've seen vines growing deep underground, well beyond the reach of 'root' plants. And I haven't ever noticed them growing back after being chopped down.
So to answer your question, no, sadly you cannot do anything to speed up the harvest of plant fibers from roots. There is no plant that you can move to increase the speed of root-growing. They simply exist, and stop existing once chopped down.
You could go to another planet and see if it has leafy trees though. Large plants (tree-sized) that have a less woody-looking construction drop a lot of plant fiber, which is great for making lots of bandages.