If there’s a plugin, it has to be found.
If not, a custom solution has to be written.
But YouTube does provide an API for that.
YouTube API v2.0 – Direct Uploading
This page explains how to upload videos to YouTube using direct
uploading. Direct uploading lets you add videos that are in your video
library to YouTube. You should choose a direct-upload implementation
if you want to host or store videos uploaded through your site and
also add those videos to YouTube. In a direct-uploading scenario, when
a user uploads a video through your site, the video will be sent to
your servers. Your application will subsequently send an API request
to upload the video from your server to YouTube.