There are several phrasings of this supposed law. The most commonly shared version is "One-armed piano players must perform for free." Websites, books, and articles proclaiming this law's existence may be found here, here, here, here, here, here, here, and here. None of these websites cite any statutes or cases.
One book quotes the supposed Iowa law as stating, "A one-armed piano player may be seen, but not if admission is charged to view his performance." This looks pretty authoritative because it is in quotation marks and sounds a bit more like it comes from a law or case. And it's in a book, not just the Internet. But, for what it's worth, websites sharing similar versions of this phrasing are here and here.
After searching through the Iowa Code and Iowa cases, I was unable to find any law or case that prohibits one-armed piano players from charging admission for their performances. The quoted, second version of the law appears in only one other website.
Based on my research of statutes and cases, I have concluded that there is no Iowa law that prohibits one-armed piano players from charging admission for their performances. As a caveat: my search was limited to the Iowa Code and Iowa case law. I did not search any municipal codes because all websites referencing this law claim that it is a state law, rather than a particular city ordinance.
If you, the reader, happen to find a particular case or statute prohibiting one-armed piano players from charging admission to their performances, I welcome you to share the citation in this post's comments. But I strongly suspect that searching for such a citation is an endeavor that is destined to fail.