This is a legal question and I am not an attorney. So, what I write here is amateur kibitzing and personal opinion.
If I was the person who wanted to use the videos of another person, I would contact that person and ask for written permission. Proper permission will include who you are, why you want to use the videos, what domain you want to post the videos on, a sample of the page where they will be displayed, how long you want to use them and any other information that might influence the owner to say "yes" and especially any information that would influence the owner to say "no". Failure to do this can get you into serious legal trouble. If the owner of the videos has done all of the legal homework, your client could be sued for $150,000 for infringement - without the owner of the videos being required to demonstrate that the infringement has caused damage.
If the owner is making money from these videos, even if it isn't serious money, he/she will probably tell anyone who wants to use them how much money must be paid for a license fee or simply tell them "no".
Really, if your client has a website about dog training, he should be making his own videos. This is best practice because it showcases the client's work, builds credibility with people who want their dogs trained, and the videos can be perfectly customized to the clients business, style, personality and location.
But, if the client is intent on using the videos of another person, he/she should get advice from an attorney who is qualified in copyright law instead of from people in an SEO forum - I sure would not go to a legal forum for SEO advice From my experience, the cost of good legal advice is a lot less than most people fear and more valuable than most people imagine.
This is all of the information that I am able to provide on this matter. Hope your client finds something that works perfectly.