The simple answer is YES!
The preferred method: All a member has to do is copy and paste the embed code from YouTube into his/her post without any additional changes or BB coding, and it will show the video for playing once the post is completed.
OR
You can create a Gallery for yourself and load the video link as a new item. Then use the bbc embed code [smg id=XX type=av] (without the quotes) in your post to add the video.
Thanks to Angel for installing the Avea mod!