This is something you probably could do. You'd swap the cold-water shutoff under the sink for one with two outputs (much better than using a saddle valve!) and run soft copper 1/4" line. Many of the shutoffs are compression fit, so this is pretty much a matter of unscrewing the compression nut, sliding the valve off, then inserting the new one and retightening (the original) nut and compression ring. You can replace those if you want, but may need a puller to remove them first. The soft copper tubing is flexible. Where are your existing pipes run? If they run in the crawl space and you haven't had problems with them freezing, then you would probably be okay. I'd worry about that.
Another option is to run it under the toekick. On new construction, they sometimes run it behind the cabinets, but you'd have to probably remove the counter on existing ones which probably isn't feasible. Depending on what is on the other side of the wall, you might be able to run the tubing there, maybe behind the moulding if you notch out the area.