Yes, 7-Eleven and FamilyMart both do, but since they're both so hard to use in English, I'll tell you about another service I found that is not tied to a convenience store chain first.
.1.Free Wi-Fi (點一點免費上網)
I haven't worked out the specifics yet but at least on the west coast at least from Lukang (鹿港) in the north to at least Kaohsiung (高雄) in the south I've very often found a free Wi-Fi signal called
I was connected to it when I wrote the first draft of this answer:
I found their logo on a phone booth in Chiayi city:
I have found these signals in big cities and small towns and perhaps even isolated villages with a convenience store, but I'm not positive yet. The signals are common but they are not everywhere. In a city you can try wandering around a street or two until you get a strong signal. In a smaller place you might not get one at all. They are not tied to convenience stores.
Usually I have to click on an image to start it working and from time to time I would get redirected to an ad in Chinese and just click on the image again:
But at one place I just got an open connection and did not have to do anything special at all for it to work.
I finally managed to find their website but it's only in Chinese and Google Translating it doesn't tell me very much. It's a product of a company called "OPEN LiFE". The Chinese name is 點一點免費上網 (diǎn-yī-diǎn miǎnfèi shàngwǎng) "little-by-little free online".
There is also a less common competitor which I've only found a couple of times, but works it much the same way. It's called YO!
7-Eleven / ibon
Although I eventually found this ad in English, I have not so far figured out how to actually sign up or log in for this free Wi-Fi in a 7-Eleven yet.
The log in page is only in Chinese and some of the text is images so even if you manage to get Google Translate to work, you still not not be able to figure out how to get it to work.
Until I found this ad I assumed it must only work with a phone sending you some code via SMS, and not with a tablet or laptop. But apparently it requires your passport and visa.
The service is available for 3 times daily, 30 minutes each. 7 days for free
FamilyMart / Famí-WiFi
Apparently FamilyMart in Taiwan also offers free Wi-Fi 3 times daily, 30 minutes each, just like 7-Eleven.
Unfortunately, not only are their sign-up and log-in pages only in Chinese, but I also can't find any kind of official word from the company in English that it exists.