Pepping studied composition at the Berliner Hochschule für Musik with Walter Gmeindl between 1922 and 1926. In 1934 he was an instructor in music theory and composition at the Spandauer Kirchenmusikschule in Berlin-Spandau, where he was appointed for much of the remainder of his life. He was given many students, for example Helmut Barbe. Pepping taught again at the Berlin Hochschule from 1935 to 1938, and from 1947 to 1968 was a professor of church music and composition there (the school was renamed in 2001 the Universität der Künste Berlin).
Pepping received for his compositions multiple tributes, an honorary doctorate from the Freie Universität Berlin in 1961 and the Kirchliche Hochschule Berlin in 1971.
He died in 1981 in Spandau.
His compositions include among others: