In this paper, we study approval voting for multi-agent knapsack problems under incomplete preference information. The agents consider the same set of feasible knapsacks, implicitly defined by a budget constraint, but they possibly diverge in the utilities they attach to items. Individual utilities being difficult to assess precisely and to compare, we collect approval statements on knapsacks from the agents with the aim of determining the optimal solution by approval voting. We first propose a search procedure based on mixed-integer programming to explore the space of utilities compatible with the known part of preferences in order to determine or approximate the set of possible approval winners. Then, we propose an incremental procedure combining preference elicitation and search in order to determine the set of approval winners without requiring the full elicitation of the agents' preferences.