The Edwardsburg conservation club has automatic throwers and so does the Saint Joseph County Restoration club, I believe both clubs run them on Sunday.
When I lived in South Bend, I always went to Chain O' Lakes Gun Club. It's $4/round for non-members, and the skeet range is open on Saturday at 1:00. Usually they close up between 4:00 and 5:00, depending on how many people are out there. They used to be open on Sundays too, but I think business is a little slower this year.
This website says Michigan City in the title, but it's actually on the west side of South Bend, just past the bypass.
It's a great place to hang out for the afternoon. The regulars are a pretty laid back group of guys. As long as you don't do anything stupid (and you help fill the machine every so often), they're happy to have new people around.