A: Will your friend come to the party tomorrow?
B: Let me ask him.
I'm having trouble translating "let" here. Should I use "lassen" as follows?
A: Wird dein Freund morgen zur Party kommen?
B: Lass mich ihn fragen.
Your option works, but is not really idiomatic. A more idiomatic thing to say would be:
Ich frag' ihn mal.
Ich muss ihn mal fragen.
mal might also be replaced by a specific time: gleich, heute abend, ... If you are B and on the phone with said friend and A throws in the question, you might also say:
Moment, ich frag' ihn mal grade.
grade indicates that you are going to do it now, you might also use this sentence when you are about to pick up your phone and call or text.
I am sure there are more possibilities, I'll add them as soon as they come to my mind again.
Your assumption is absolutely correct!
Of course you can also use something like
Ich muss ihn erst fragen.