Why isn't there an apostrophe of possession?

No. The noun phrase 'U.S. Department of Education' is used attributively — which means 'like an adjective'.

Nouns are often used like this:

Our school canteen.
The car doors.
A shop window.

Jonathan1Shouldn't it be written as "A recent U.S. Department of Education's report..."?

No. Besides what was written above, note also that you can't have "a" and a possessive together. For example, a John's friend, a Mary's dress, and a Department of War's fighter jet are all completely wrong grammatically.

Jonathan1Why isn't there an apostrophe of possession after the 'Education'?

Because genitive noun phrases (normally) function as determiners, and report already has a as its determiner. There can only be one determiner per noun phrase:

a recent U.S. Department of Education report .......

Thank you all for your answers!
So just to be sure, if there is an 'a', there will never be an 's' that represents prossession?

Jonathan1So just to be sure, if there is an 'a', there will never be an 's' that represents prossession?

Not normally, but it’s possible for a genitive to have attributive rather than determiner function, as in the common phrases an old people’s home, a women’s college, a bachelor’s degree.

Also "a day's pay" and "a hard day's night".

Oops. I answered the last question all wrong. It’s quite common to have a(n) + genitive noun when they form a noun phrase together. Not sure why I was only thinking of a(n) + attributive genitive.

Thank you for your responses.