http://bugs.winehq.org/show_bug.cgi?id=31159
--- Comment #3 from Juan Lang juan.lang@gmail.com 2012-07-09 10:46:24 CDT --- The problem seems to be: trace:chain:CRYPT_BuildSimpleChain Couldn't find issuer, halting chain creation
You're connecting to https://store.eun1.lol.riotgames.com, which has the chain: GTE CyberTrust Global Root DigiCert High Assurance EV Root CA DigiCert High Assurance CA-3 store.eun1.lol.riotgames.com
According to your log, when using builtin wininet, the GTE CyberTrust Global Root certificate can't be found when verifying the chain for store.eun1.lol.riotgames.com. It's curious that using native wininet works around it, since that also uses builtin crypt32.
I'll try to have a closer look when I get a chance.