Mobile Edge Computing (MEC) has emerged as a promising technology to push the cloud frontier to the network edge, provisioning network services in the proximity of mobile users. Serving mobile users at the edge of the service network can reduce service latency, lower operational cost, and improve network resource availability. Along with MEC technology, Network Function Virtualization (NFV) is another promising technique that implements various network functions as pieces of software in servers or clusters of servers. Providing virtualized network service in MEC can improve user service experience, simplify network service deployment, and ease network resource management. However, mobile users usually move in networks arbitrarily, and different users demand different services with differentiate delay requirements. In this paper, we study mobile users requesting for virtualized network function services in MEC by formulating two novel user request admission problems that take into account user mobility and service delay requirements. We first devise an efficient approximation algorithm with a provable approximation ratio for the network utility maximization problem. We then develop an efficient online algorithm for the online throughput maximization problem. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising.