You can use the embed code on YouTube's site, and make a custom BB code so users can display YouTube movies inside of their posts.

Admin CP -> Custom BB Codes -> Add New BB Code

Title - YouTube
Tag - youtube

Replacement code:
<table class="tborder" cellpadding="6" cellspacing="1" border="0" width="466"><tr><td class="smallfont"><strong>IMPORTANT NOTICE:</strong> No media files are hosted on these forums. By clicking the link below you agree to view content from an external website. We can not be held responsible for the suitability or legality of this material. If the video does not play, wait a minute or try again later.</div></td></tr>
<tr><td class="smallfont" align="right"><a href="#" onclick="return toggle_collapse('{param}')"></a><a href="#" onclick="return toggle_collapse('{param}')"> I AGREE</a></td></tr><tbody id="collapseobj_{param}" style="display:none;"><tr><td class="alt1" align="center"><object width="425" height="350"><param name="movie" value="<A href="http://www.youtube.com/v/{param}"></param><embed">http://www.youtube.com/v/{param}"></param><embed src="http://www.youtube.com/v/{param (http://www.youtube.com/v/%7Bparam)}" type="application/x-shockwave-flash" width="425" height="350"></embed></object></td></tr></tbody>
</table>

Example Usage - JFwCCL0Vh6U

Please add a screenshot..