1.They were waiting at the airport all morning.
This is the normal natural way to say it. You don't need to use the definite article here, or with "all afternoon", "all day", "all night").
You could say,
They were waiting at the airport all morning long.
2. They were waiting at the airport all the morning. This is possible if you are referring to a particular morning or just as a matter of style.
They were waiting at the airport all the morning of the fifteenth of June.
They were waiting at the aiport the entire morning.
They were waiting at the airport all morning on the fifteenth of June. (again without "the")
"all the morning" is possible in poetry or song.
From The Gardener
By Rabindranath Tagore
I try to wreath all the morning, but the flowers slip and they drop out.
You sit there watching me in secret through the corner of your prying eyes.
Ask those eyes, darkly planning mischief, whose fault it was.
I am from a land far, far away, across the sea.