Excitation of chaotic magnetization dynamics in nanomagnets is of great interest because it bridges the condensed matter physics and nonlinear science and has a potential to emerging technologies such as neuromorphic computing. However, it has been difficult to observe and identify chaos in spintronics devices because the excitation of chaos requires dynamics in a large-dimensional phase space, according to the Poincaré-Bendixson theorem. An efficient way to overcome this issue is using feedback, which enables the dynamical degrees of freedom to be increased even in a single device. Here, we experimentally demonstrate the excitation of chaos in a vortex spin-torque oscillator by utilizing a feedback circuit. The radio-frequency current emitted by the oscillator flows in the feedback circuit and is converted into an oscillating magnetic field. The oscillating field generates a torque acting on the vortex and modulates its dynamics, resulting in chaotic dynamics which can be tuned by electrical means.