It is necessary that an efficient worker__________his work on time.

选accomplish.
本句句首的It作形式主语,真正的主语是that引导的从句,即:that an efficient worker ... on time.that引导的主语从句采用了虚拟语气,其完整的形式为It is necessary that an efficient worker (should) accomplish his work on time.
本句省略了主句从句中的should,因此accomplish为正确答案.
PLUS: 虚拟语气与不定式
1.It is necessary + that ... 的结构中,that引导的主语从句的谓语动词要用虚拟语气,即 should + 动词原形,其中的should 可省略.
e.g. It is necessary that we (should) finish the task tomorrow.
2. 在不定式作形容词的主语的句式中,It is necessary + 不定式
e.g. It is necessary for you to lock the car when you do not use it.