Can this be cooked on the stove? My oven doesn't work.

You can braise this on the top of the stove. You will need a heavy dutch oven with a snug fit lid, and be able to maintain a low simmer for a long time. Instead of transferring it to the oven, you will just be bringing it to a simmer on the stove. Pop on the lid, and continue with the basting as directed. You will probably need to remove the lid for the last half hour or so of cooking so the sauce actually reduces more. If you lid doesn't fit snugly, you may also need to add more liquid. What counts is that the chicken cooks slowly in the sauce, which is what braising is all about. An oven just helps keep the temperature even all around the pan- and the stovetop can be a little more fiddly and attention demanding. But braising was done over a flame for a lot longer than it's been done in an oven, so it can certainly be done in modern times.