Many musicians learn to overcome various handicaps (including missing fingers) and become very accomplished on their instrument of choice. People play with their head and their heart - their fingers are just a means to execute what they wish to express musically. Given enough desire and commitment, anyone can learn to play piano or guitar with fewer than ten fingers.
Django Reinhardt was a famous jazz guitarist who lost the use of two of his four fretting fingers but learned to adapt and went on to become one of the most accomplished and talented guitarist in the world.
Malcolm John “Mac” Rebennack, Jr. better known as Dr. John decided to switch from guitar to piano after losing his index finger on his fretting hand in an accident. As a pianist he became a highly sought after professional session musician and his work on piano can be heard on albums by James Taylor, Van Morrison and Aretha Franklin and has worked with Mick Jagger, Eric Clapton, Willie Nelson, and Ani DiFranco, and was inducted into the Rock and Roll Hall of Fame in 2011.
According to Wikipedia
Paul Wittgenstein (May 11, 1887 – March 3, 1961) was an Austrian concert pianist notable for commissioning new piano concerti for the left hand alone, following the amputation of his right arm during the First World War. He devised novel techniques, including pedal and hand-movement combinations, that allowed him to play chords previously regarded as impossible for a five-fingered pianist.
Paul Wittgenstein commissioned numerous works from famous composers (including Benjamin Britten, Paul Hindemith, Alexandre Tansman, Erich Wolfgang Korngold, Sergei Prokofiev, Franz Schmidt, Sergei Bortkiewicz, and Richard Strauss) designed to be played on piano with only the left hand.
Here is a You Tube video about a very talented pianist with only 4 fingers.
YouTube 4 fingered pianist
There are countless other examples and YouTube videos of musicians who have lost fingers who are able to play piano or guitar amazingly well. This proves my opening point that given enough desire, anyone can learn to overcome their handicap in a way that allows them to pursue their passion and express themselves musically.
As far as piano pieces strictly arranged for a specific handicap, I am sure they exists, but each situation would be different depending on how many fingers are missing and on which hand.
Here is a link that provides a list of music composed specifically for left hand only piano (many commissioned by the aforementioned Paul Wittgenstein).
List of Works for left hand piano
It would probably be a good idea for an aspiring pianist who was deprived of the use of one or more fingers to find a patient piano teacher who could help them discover the appropriate alternative fingerings for music they wanted to play.