I need someone who can make me

Yours is correct, and you can also say, "I am looking for someone who can make me laugh."

Need (=require)

1. need somebody/something
Your example is correct.

2. need somebody to do something
e.g. I need someone to make me laugh.

We seldom say, "I need somebody for making me laugh."

Your version is correct. In American slang, you can say "I need someone who can crack me up."

這句句子的文法沒有錯.

As a modal verb,and participle;
I need someone for making me laugh.