'inquire why' is correct and usable in written English.
You can use it when you would like to ask someone the reason for something. For example, "I was asked to inquire why the payment was late."
The board at least has a duty to inquire why this decision is being made.
I got on the phone to Faber, to inquire why he didn't simply tell me we had a problem.
A lady wrote to the publishers of the Oxford English Dictionary to inquire why every word starts out with a capital She said this confused her.
Instead of snatching hats, as he is wont to do, George would simply gaze at the man with the yellow hat and incessantly inquire, "Why is it yellow?
A second question, then, might inquire why the great tycoon allowed Wolff to sit there waving a microphone in his face for day after supposedly busy day?
"Learning, no," was his disarming reply, making me inquire why he didn't spend more time composing if he knew all there was to know about these pieces.
We inquired why she was serving tea there.
CNN's weekly press critique, "Reliable Sources," inquired why.
Another character innocently inquires, "Why'd you dress so glocky, miss?
Whenever I inquired why, I was told that he is a peerless professional.
When he inquired why he did this, the old man replied that he could not afford a slave.
