Vehicles require timely channel conditions to determine the base station (BS) to communicate with, but it is costly to estimate the fast-fading mmWave channels frequently. Without additional channel estimations, the proposed Distributed Kernelized Upper Confidence Bound (DK-UCB) algorithm estimates the current instantaneous transmission rates utilizing past contexts, such as the vehicle's location and velocity, along with past instantaneous transmission rates. To capture the nonlinear mapping from a context to the instantaneous transmission rate, DK-UCB maps a context into the reproducing kernel Hilbert space (RKHS) where a linear mapping becomes observable. To improve estimation accuracy, we propose a novel kernel function in RKHS which incorporates the propagation characteristics of the mmWave signals. Moreover, DK-UCB encourages a vehicle to share necessary information when it has conducted significant explorations, which speeds up the learning process while maintaining affordable communication costs.