We consider the problem of designing controllers for nonholonomic mobile robots converging to the source (minimum) of a field. In addition to the mobility constraints posed by the nonholonomic dynamics, we assume that the field is completely unknown to the robot and the robot has no knowledge of its own position. Furthermore, the field is randomly switching. In this paper, we combine ideas from stochastic approximations and nonholonomic control, in order to address this challenging problem. In particular, we develop a rotation-invariant and forward-sided version of the simultaneous-perturbation stochastic algorithm, which is much more suitable for sensor-free navigation. Based on this algorithm, we design source seeking controllers for nonholonomic robots and prove convergence to the unknown source with probability 1. The proposed controllers are demonstrated by numerical simulations.