。
She requested leave from the section chief.
Literal
She [topic-は] section-chief [to-に] leave [object-を] applied-for.
申し出る (もうしでる) means to propose, offer, or request — a polite/formal verb built from 申す (humble for 言う) + 出る (to come out). 休暇を申し出た = requested time off. This is more formal than 休みをもらう or 休みを取る, reflecting the workplace hierarchy: you formally request leave from a superior. 主任 (しゅにん) is a section chief or team leader — typically a junior rank in the management chain, below 課長 (section manager) and 部長 (department head).