Mobile Edge Computing (MEC) has recently emerged as a promising technology to push the cloud frontier to the network edge, provisioning network services in close proximity of mobile users. Serving users at edge clouds has many advantages, such as reducing service delay, lower operational cost, and improved network resource availability. Furthermore, providing virtualized network service in MEC can improve user service experience, simplify network service deployments, and ease network resource management. However, provisioning reliable and seamless virtualized network services for mobile users while meeting their individual stringent delay requirements is of significant importance and challenging. In this paper, we study mobile users requesting for virtualized network function services in MEC. We first formulate two novel user request admission problems that take into account user mobility and service delay requirements. One is to minimize the admission cost of all user requests, assuming that there are sufficient resources in MEC to meet user resource demands; the other is to maximize the accumulative network utility, subject to resource capacities of cloudlets, where the utility gain by admitting a request is determined by its resource demand and delay requirement, and the requested resource utilization in MEC. We then devise efficient approximation algorithms for the two problems. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising.