Fetching details from the dictionary...

to apply for; to make an application; to propose (marriage); to offer (mediation); to make an overture (of peace); to challenge; to lodge (objections); to request (an interview); to subscribe for; to book; to reserve
大学の入学試験に申し込んだ。
I applied for the university entrance exam.
新しい仕事に申し込むつもりです。
I intend to apply for a new job.
図書館のカードを申し込んでいる。
I applied for a library card.