Janice, I am not an expert, but I have done a little research assisting my dad, who wrote a book about his service in the Marine Corps in WWII.
At one time he wanted to check who was present and accounted for in his unit during part of a campaign in the South Pacific. We had success doing so using the following contact info and request format. We wrote to:
Reference Section Marine Corps Historical Center Building 58 Washington Navy Yard Washington, DC20374
If you would be so kind, please send me a Muster Roll of Officers and Enlisted Men for the Second Battalion, Fifth Marines, First Marine Division, Fleet Marine Force for the month of October, 1944.
I can't remember how I acquired this contact info. Neither can I tell you if the Corps had "morning reports", or if they had them and called them something else.
However, this might be a good place for you to start if you don't have other resources available.