While the Holy Grail was always depicted as an important or powerful relic, in medieval literature I have never seen a quest to own the Grail, only to seek it in a spiritual sense. In early stories, the seeking of the Grail was subordinate to a quest to fix a mistake. This can be seen in Perceval, by Chretien de Troyes (c. 1190), and in its adaptation Parzival, by Wolfram von Eschenbach (c. 1210).
Later, the Grail became the actual object of the quest. This is seen in the Vulgate and Post-Vulgate versions of the Matter of Britain, most noticeably in Malory's Le Mort D'Arthur (1485). In these later works, the objective of the quest was to be worthy of the Grail and to serve it (for instance, taking it from Corbenic to Sarras), and never to own it.
In most recent works, a search for the Grail often has its possession as objective, due to its powers or significance. The earliest modern examples I know of are War in Heaven, by Charles Williams (1930), and Wagner's opera Parsifal (1882), in which Klingsor, having the Spear, also wants the Grail in order to destroy Amfortas. Even in The Once and Future King (1958), an Arthurian romance by T. H. White, it is initially implied that the objective of the quest is to bring the Grail to Arthur. As far as I know, this does not happen in the medieval romances. Examples in cinema are Indiana Jones and the Last Crusade (1989) and the loose adaptation of Arthurian romances Excalibur (1981).
My question is: when did literature start to depict the Holy Grail as an object to be owned? I do not restrict the question to works that present the Grail as the Holy Chalice.