Tai Nam is not on Argyle. It's in "Tai Nam Market Center", about a half-dozen stores south of Argyle right off Broadway on the right (east) side, and which is a "parking lot" with a strip mall on the south side of that parking lot with Tai Nam supermarket/grocery at the east end of that strip.
I have always seen kaffir lime leaves there whenever I've been there.
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I have gotten kaffir lime leaves from Ted Biernacki at the OP Farmers' Mkt. He does not always have them, so call him first.
Ted's Greenhouse in Tinley Park:
16930 S 84th Ave
Ted is at several Farmers' Markets during the summer:
Hinsdale Farmers Market
30 E Chicago Avenue
At the intersection of Garfield Street and Chicago Avenue - north of the Brush Hill Train Depot.
Monday 7:00 AM - 12:00 PM
Hines V.A. Hospital (by Loyola Hospital
)5th Avenue & Roosevelt Rd.
Thursday 7am - 1pm
Oak Park Farmers' Market
SW Corner of Pilgrilm Church parking lot
640 W Lake St (1 blk w of Ridgeland)
Sat. 7am - 1pm
He asked me one time if I wanted to just buy a tree so I'd always have the leaves, so I guess he has them available. Said they're 'easy to grow'. And that the leaves freeze well.
Don't miss the donuts & folk music at Oak Park!
They're hard to find. Sometimes they are in the freezer. Those work quite well. I was thrilled to find them fresh at Assi in Niles, a suburb northwest of Chicago. I always ask the employees. They often love to talk about their cuisine and they know you understand it somewhat if you are asking for kaffir lime leaves. I always ask about their ideas for a curry without coconut milk that is sometimes called Jungle Curry, Kang Pa, Gang Par.....The hottest curry I know of and so good!
Availability comes in spurts, and I think there are legal reasons for this. Sometimes they are banned from being imported (iirc.) My usual spot is the Asian grocery store on Broadway a block or so N of Lawrence in the strip mall on the East side of Broadway. It's in the NE corner of that strip mall. My other "go-to" spot for these is Super H Mart up in Niles.