In wireless sensor networks, the data collected by sensors are usually forwarded to the sink through multi-hop forwarding. However, multi-hop forwarding can be inefficient due to the energy hole problem and high communications overhead. Moreover, when the monitored area is large and the number of sensors is small, sensors cannot send the data via multi-hop forwarding due to the lack of network connectivity. In order to address those problems of multi-hop forwarding, in this paper, we consider a data collection scheme that uses mobile data collectors (MDCs), which visit sensors and collect data from them. Due to the recent breakthroughs in wireless power transfer technology, MDCs can also be used to recharge the sensors to keep them from draining their energy. In MDC-based data-gathering schemes, a big challenge is how to find the MDCs’ traveling paths in a balanced way, such that their energy consumption is minimized and the packet-delay constraint is satisfied. Therefore, in this paper, we aim at finding the MDCs’ paths, taking energy efficiency and delay constraints into account. We first define an optimization problem, named the delay-constrained energy minimization (DCEM) problem, to find the paths for MDCs. An integer linear programming problem is formulated to find the optimal solution. We also propose a two-phase path-selection algorithm to efficiently solve the DCEM problem. Simulations are performed to compare the performance of the proposed algorithms with two heuristics algorithms for the vehicle routing problem under various scenarios. The simulation results show that the proposed algorithms can outperform existing algorithms in terms of energy efficiency and packet delay.