The vehicle routing problem (VRP) is an NP-hard optimization problem that has been an interest of research for decades in science and industry. The objective is to plan routes of vehicles to deliver goods to a fixed number of customers with optimal efficiency. Classical tools and methods provide good approximations to reach the optimal global solution. Quantum computing and quantum machine learning provide a new approach to solving combinatorial optimization of problems faster due to inherent speedups of quantum effects. Many solutions of VRP are offered across different quantum computing platforms using hybrid algorithms, such as quantum approximate optimization algorithm and quadratic unconstrained binary optimization. In this work, we build a basic VRP solver for three and four cities using the variational quantum eigensolver on a fixed ansatz. The work is further extended to evaluate the robustness of the solution in several examples of noisy quantum channels. We find that the performance of the quantum algorithm depends heavily on what noise model is used. In general, noise is detrimental, but not equally so among different noise sources.