When I used the "Attach files" button, I selected the Photo Library, then went to "Videos" and checked-marked a video, hit "Done", and waited for it to reach 100% on the upload. Then it gave me this error message:
So... I guess I would have to convert a video to a .gif first, before it would let me upload direct from my phone.
Or, upload it to a YouTube account and share the link or embed it like any other "approved" video source.
If you have a Facebook account you might try messaging the video to yourself, and then download it on your computer; if that is where you are wanting to get it to.
Emailing to yourself may work to to get it over, but many email sites have file size restrictions I have found.