Client
noun
You already know that 依頼 means "request." What's a "request" person? It's someone who requests things from you. In other words, your client! 依頼人 generally refers to someone requesting particular services or assistance. Whether it's consulting a lawyer, employing a detective, or initiating a bank transfer, they're the one making the request.
いらいにん
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
もう少し我慢すれば、依頼人が誰だか分かったのに。
If you had a little more patience, you could have found out who the client was.