You would need to keep a database of YouTube video links before you can start playing them at random. To create a script which extracts all the video links from YouTube is a very hard task. You would be better creating a database of links manually and getting php to display one at random based on the seek.
My $0.02
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or you could just use a script to search and scrape youtube links for certain strings like "hilarious" or whatever tag you want it to search and display a video for.
What kind of price are you trying to pay for a custom script?
Well, i can develop that for you. If you check my software, SteadyNiche, (look at my signature), you can see that there's a video section. I can do something like that and even more.
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I know that there is one out there... But I just cannot find it. I've seen it about a week ago. I'm not sure if it was random or random from tags related to your website.