I've never heard of trading in a winning lottery receipt for goods at a 7-11. I've won 200 NT several times, and I have always cashed in the receipts at a bank. And it can be any bank.
You have to fill in a "Taiwan ID number" (shen1 fen4 zheng4 zi4 hao4) on the back of the receipt. Until about a year ago, foreigners in Taiwan didn't have an ID number, so I filled in my wife's ID number and took her ID card to prove it. (She didn't have to go in person.)
But since about a year ago, foreigners who have an ARC card can get an ID number from the Foreign Affairs Police and they write it on the back of your ARC card. But the format is slightly different than for Taiwanese people. For Taiwanese people, the ID number is one letter followed by 9 numbers, but for foreigners, the format is two letters followed by 8 numbers.
I haven't won any lottery receipts since I got my new ID number, so I'm not 100% sure about this, but I think the banks will probably let foreigners use their ID numbers to cash in lottery receipts because for everything else that has required a Taiwan ID number (e.g. going to hospitals, opening up a new bank account, applying for a new phone line, setting up accounts with the power company, the water company, and the cable TV company, etc.), I have had no problems at all using the ID number that the Foreign Affairs Police gave me whenever anyone asks for my "shen1 fen4 zheng4 zi4 hao4".