Discourse does support video uploads:
Out of the box this is controlled by 3 site settings:
authorized_extensions_for_staff
theme_authorized_extensions
and
authorized_extensions
Out-of the-box video uploads are not included in authorized extensions. To add them add the appropriate extensions. Eg: mp4 and other video containers.
Also be sure to allow for larger uploads by adjusting max_attachment_size_kb.
Keep in mind that low trust users (eg: trust level 0) can not post videos.