If you are selling a product that includes potentially copyrighted work is an infringement on the copyright owner's rights. You may potentially be sued. The best way to go about it is contacting the artists or the production houses owning the copyright and request permission. Please make sure you tell them you are planning to sell your DVDs. You would probably need a lawyer to draft the permission forms, and other documents you may need, etc.
This answer is provided for general education purposes only and is not intended to provide, nor does it provide, any legal advice. By viewing this answer you understand and expressly agree that there is no attorney-client relationship between you and the attorney who authored the answer. Should you need legal advice, please contact a licensed attorney who practices in this area. Readers of this answer and the information contained herein should not act upon any information contained in this answer without seeking legal counsel.
If it is a live show put onto a DVD this will probably have protected expression under copyright law. Taking music from another and incorporating it into the puppet show is tricky. Music under copyright law potentially carries several copyright issues and not just one as some believe. The copyright protection in music may well have several contributors.
It may be the case that you will probably have to contact several different people or institutions. And yes, U.S. copyright law probably protects the music by the artist. Your best bet is to consult with an attorney that has experience with intellectual property law and similar matters. This attorney should have a good understanding of the complexities of your matter and can provide a legal consultation and analysis of your situation. This communication does not constitute legal advice, and is for informational purposes only.
In order to use someone elses work (such as a song) you need to obtain various rights and clearances.
The first thing to do is to find out who the publisher of the music is.
This can be found online with a search of ascap, bmi or thru the US Copyright office.
Contact the publisher and send them a letter with the name of the song; the length of the portion of the song you wish to use, how you wish to use it and what you expected sales and revenue will be.
Many publisher will grant a Gratis license for charitable purposes.
You have to ask.
If you are attaching this to a DVD you can ask the publisher for the contact person for mechanical rights as well.
Get everything in writing in case you are ever sued