We consider a class of submodular maximization problems in which decision-makers have limited access to the objective function. We explore scenarios where the decision-maker has access to only k-wise information about the objective function; that is, they can evaluate the submodular objective function on sets of size at most k. We begin with a negative result that no algorithm using only k-wise information can guarantee performance better than k/n, where n is the size of the selected set. We present an algorithm that utilizes only k-wise information about the function and characterizes its performance relative to the optimal, which depends on a new notion of curvature of the submodular function. Finally, we present an experiment in maximum entropy sampling that highlight the approximation performance of our proposed algorithm.