I'm afraid it's not possible at the moment. Web pages can't directly access files on disk for security reasons, and currently there's no other API for web pages to register a file type association. The desktop build can do it if you manually associate it with .c3p files.
Could an official C3 browser extension solve problems like these?
I assume that those would have more permissions (e.g. local file access), since they require to be installed manually.
(Also created this topic regarding browser extensions to workaround permission issues quite a while ago.)