First of all, I would recommend playing on modes and maps that easy to win on for you. For example, when I play on toy factory in team fight, I always get 5 or 4 ribbons. Just farm and when you have won 5/6 times, you'll be able to open a chest.
Second, use gems to open the lucky chest. You can get gems from the free gems and levelling up. I've gotten ribbons from the lucky chest several times and they usually give 25, and rarely 10.
These are the ways I use to farm for ribbons, and I hope you get good loot from your chests.