No, actually, it's not legal. Those video clips you're talking about are pirated excerpts from Yngwie's instructional video/DVD which is copyrighted. I would suggest buying the DVD if you really want to learn the licks, but I don't think it's worth it. You'd be better off getting a better video -- Yngwie's just shows a couple licks, no real theory behind how he creates the licks. I'll transcribe one lick I learned from my DVD, which I think is the most commonly available on the net... it's an A Minor run, and all played on the high e string.

|-8-5-7-8-10-7-8-10-12-8-10-12-13-10-12-13-15-12-13-15-17-|

It's been a while since I've seen the video, so that second to last note may be a G#, and not a G, because it's pretty characteristic of Malmsteen to use the Harmonic Minor in his licks.