We present a new optimization strategy for small-to-intermediate scale variational quantum algorithms (VQAs) on noisy near-term quantum processors which uses a Gaussian process surrogate model equipped with a classically-evaluated quantum kernel. VQAs are typically optimized using gradient-based approaches however these are difficult to implement on current noisy devices, requiring large numbers of objective function evaluations. Our approach shifts this computational burden onto the classical optimizer component of these hybrid algorithms, greatly reducing the number of quantum circuit evaluations required from the quantum processor. We focus on the variational quantum eigensolver (VQE) algorithm and demonstrate numerically that these surrogate models are particularly well suited to the algorithm's objective function. Next, we apply these models to both noiseless and noisy VQE simulations and show that they exhibit better performance than widely-used classical kernels in terms of final accuracy and convergence speed. Compared to the typically-used stochastic gradient-descent approach to VQAs, our quantum kernel-based approach is found to consistently achieve significantly higher accuracy while requiring less than an order of magnitude fewer quantum circuit executions. We analyze the performance of the quantum kernel-based models in terms of the kernels' induced feature spaces and explicitly construct their feature maps. Finally, we describe a scheme for approximating the best-performing quantum kernel using a classically-efficient tensor network representation of its input state and so provide a pathway for scaling this strategy to larger systems.