Why aren't .webm files allowed on Boards?

Gifs and other files are not "uploaded" to the boards, they are loaded from other sites. Given Riot's player base size (aka, massive), even a moderate amount of clip posting would eat up tons (or tonnes for Canadians & Brits) of storage, not to mention the bandwidth it would take to serve those files.

Then there's the regulatory ramifications (think DMCA in the US, and similar copyright laws in the EU...which recently became more strict).

Basically, it would be a mess, and the benefit would not be worth the effort.

Nothing is uploaded directly to the boards. Even images are hosted off-site.

Mainly because the boards do not host anything but text and the built-in graphics. Anything else is hosted off-site because, quite simply, the boards do not have an upload or storage function for user-uploaded content.