Stackelberg games have been widely used to model interactive decision-making problems in a variety of domains such as energy systems, transportation, cybersecurity, and human-robot interaction. However, existing algorithms for solving Stackelberg games often require knowledge of the follower's cost function or learning dynamics and may also require the follower to provide an exact best response, which can be difficult to obtain in practice. To circumvent this difficulty, we develop an algorithm that does not require knowledge of the follower's cost function or an exact best response, making it more applicable to real-world scenarios. Specifically, our algorithm only requires the follower to provide an approximately optimal action in response to the leader's action. The inexact best response is used in computing an approximate gradient of the leader's objective function, with which zeroth-order bilevel optimization can be applied to obtain an optimal action for the leader. Our algorithm is proved to converge at a linear rate to a neighborhood of the optimal point when the leader's cost function under the follower's best response is strongly convex and smooth.