I haven’t done this, yet, but did some rigging on Napoleonics ships from Warlord Games where I had very good results with thin yarn. Make a loop, place it around the horse’s head, fix it with some super glue, then either wrap it around the hands of the driver or do the loop trick again, fix it with superglue. If you want the strings tight, you can use superglue, here, too.
Cut off any yarn with small scissors.
I’m really no pro, but this was my second or third attempt. I used the very thin nylon yarn in the Black Seas box sold by Warlord Games which was my Xmas present some years ago.