Having the patches sewn on is the best option if you don't want to move the patches around. If you do, you'll need something that sticks to leather and the patch. Double sided tape? Hmm, I doubt that would be secure enough. You might try something like
this iron on adhesive fabric for a better hold. I don't know how removable/reusable it is for moving patches around though.
Velcro seems like a good solution for keeping the patches mobile. Would you be interested in putting velcro all over your leather vest? In my opinion, it would ruin the look of the vest, but to each his own. You could also try small safety pins. It would leave small holes so keep that in mind.