Fetching details from the dictionary...

to be fit (for a role); to be qualified (for); to be equal (to a task); to be able to fulfill one's duties (as)
この責任ある仕事が自分に務まるか不安に感じている。
I'm anxious whether I'm competent for this responsible job.
新人ながら部長の代理が十分務まった。
Despite being new, he was fully qualified to act as the department head's substitute.
経験が浅くても、彼にはその役割が務まっている。
Even with limited experience, he is fulfilling that role capably.