After studying law, archaeology and art history at the University of Innsbruck, Vinzenz Oberhammer habilitated in 1936 in art history. On 1 April 1938 he became curator at the Tiroler Landesmuseum Ferdinandeum, which had often consulted him hitherto as an expert. As a curator who also served several times in the Wehrmacht, he was responsible for the collections, for assessing the art-historical relevance of planned acquisitions, dealing with organizational matters and designing and carrying out exhibitions. Together with Oswald Trapp, chairman of the Museumsverein from 1937 to 1945, he prepared the agendas and kept the minutes of the committee meetings. The Ferdinandeum hoped to be able to acquire artworks from many seized collections. Oberhammer made the selection and sent his requests to the Zentralstelle für Denkmalschutz (Central Monument Protection Office). The first objects from Oskar Bondy's collection were delivered in August 1940. As Trapp functioned in subsequent years as Gaukonservator in the culture commission in South Tyrol, Oberhammer organized the acquisitions. He was particularly interested in purchasing glasses from Franz Ruhmann's collection and porcelain ware from Nathan Eidinger's assets. When he was not serving in the Wehrmacht, he was also responsible from 1942 to 1945 for the storage of the Tiroler Landesmuseum holdings. On 17 June 1939, Oberhammer had applied for NSDAP membership, which had been granted to him on 1 January 1941 with the number 9,672,651.

After the war he was involved with museum chairman Ernst Durig (1870–1965) in the restitution of artworks acquired during the Nazi period, including objects he had himself been responsible for, from the following collections: Bernhard Altmann (1949, 1950), Gustav Arens (1948), Oskar Bondy (1947), Nathan Eidinger (1953), David Goldmann (1948), Felix Kornfeld (1950), Ernst Pollack (1948), Albert Pollak (1948), Robert Pollak (1947), Friedrich Reitlinger (1945, 1946, 1947), Alphonse Rothschild (1947), Louis Rothschild (1946, 1947), Franz Ruhmann (1951), Leontine Samek (1948), Paul Schwarzkopf (1948–1957) and Alfons Thorsch (1947). The reopening in 1949 of the museum, which had been damaged in April 1945, the reorganization of the collections in 1950 and many exhibitions took place during Oberhammer's time as curator of the Ferdinandeum. He was appointed in 1955 to the Kunsthistorisches Museum in Vienna, where he succeeded Ernst Buschbeck as head of the painting collection, and was also administrative director. In 1960 he assumed the office of First Director, which he held until his retirement on 31 December 1966