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What are the solutions in C2 to create an IAP system for web browser games?
There are many threads regarding this, try running a search. Have you checked in the stickied FAQ thread? how-do-i-frequently-asked-questions_t63692 - Specifically in the section "Monetizing".
Try also https://www.scirra.com/tutorials/4848/h ... -crosswalk
EDIT: My apologies, missed the web browser game part >< See below.
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Short answer: There's none, short of finding a payment handling service that has an api, and writing a plug.