Copyright issue

Technically....there's no copyright if the work isn't legally copyrighted and there's no trademark violation if it's not a registered trademark or (logo).   But as Warren said taking someone else's logo poses inherent problems and is just a lame thing to do.

I spend a good deal of time searching for commercially usable images and I can't tell you how often I see people who take other's people work that the original author has made freely available and then license it under an attribution license.  So lame.

Just create an original logo or even mash something up.   Use words or even simple geometric shapes which

The fact that you would even consider just straight up taking someone else's logo and publishing it as your

own (and probably even slapping a license it) is.....LAME.  I'll be the bad guy & say it.

Now I'll say something less helpful to you then what I've already said

Do a google image search then click Tools>Usage Rights> Reuse with modification 

That will filter the results to only usable images.

Also there are sites like:

https://pixabay.com/

Internet Archive: Digital Library of Free Books, Movies, Music & Wayback Machine​

https://www.loc.gov/collections/

National Archives |

Foter: Free Stock Photos & Images

You still need to check the licenses an all except for Pixabay & National Archives

There's an inherent problem with using someone else's logo: viewers will think that the person or company that the logo represents made the video.  Your friend should provide you with his or her logo to animate (or create one if it does not already exist).