Can someone tell me the proper way to punctuate this sentence:

-- Respectfully asking why. --

The intent is more or less for a tagline of sorts. It's meant to come across that the subject isn't afraid to ask why but would always do it in a respectful manner.

This is just my opinion; others will probably disagree.
The "proper" way would be:
-- Respectfully asking, "why?" --
Most people nowadays would omit the comma. I think it is recommended because of the change to quoted speech, but commas normally indicate a pause and most people would not pause there.
Personally, in informal circumstances in a message signature, I would just use your original sentence, with the full stop removed:
-- Respectfully asking why --
The phrase "to ask why" is easily recognised and has a well-understood meaning, and that is why I wouldn't use the quotes and question mark - simply because they're unnecessary. I think the phrase is parsed more quickly and easily without them. It's rather like an advertising slogan. These are seldom grammatically correct or complete - remember Apple's "Think different"?
Some here will probably disagree, seeing this simplification as sloppy and incorrect.
KrisBlueNZ-- Respectfully asking, "why?" --
I completely agree with you and how I've had it all along but I'm just not 100% sure on this one. I left all punctuation out of the sentence in question so it wouldn't sway anyone's opinion.

Thank you.

If anyone else wants chime in with approval or disapproval, that would be super. I still have another day before it has to be finalized.

Thanks everyone.