Vehicle routing problems (VRPs) can be solved as optimization problems. Practical applications of the VRPs are involved in various areas including manufacturing, supply chain, and tourism. Conventional approaches using von Neumann computers obtain good approximate solutions to the optimization problems, but conventional approaches show disadvantages of computation costs in large-scale or complex problems due to the combinatorial explosion. Oppositely, Ising machines or quantum annealing machines are non-von Neumann computers that are designed to solve complex optimization problems. In this paper, we propose an Ising-machine based approach for the vehicle routing problem with balanced pick-up (VRPBP). The development of the VRPBP is motivated by postal items pick-up services in the real-world. Our approach includes various features of VRP variants. We propose a 2-phase approach to solve the VRPBP and key elements in each phase are mapped onto quadratic unconstrained binary optimization (QUBO) forms. Specifically, the first phase belongs to the clustering phase which is an extension to the knapsack problem with additional distance and load balancing concerns. The second phase is mapped to the traveling salesman problem. Experimental results of our approach are evaluated in terms of solution quality and computation time compared with conventional approaches.