At its core in TOEICer's you have one TOEICer, and in TOEICers' you have multiple.
TOEICer's club focuses on the individual. "Are you doing the TOEIC? This is the club for you!" sort of thing. TOEICers' club focuses on multiple people; "This is a club for all the people doing the TOEIC." TOEICer club is similar to the first, but without the apostrophe -- that's just a matter of style.
You'll see "Mother's Day" and "Mothers' Day" both. The former with a feeling of "A Day For My Mother" and the latter with a feeling of "A Day For Mothers Around The World". It's a subtle distinction, at any rate.
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