Does the sentence
It requires full orchestra and progressively adds each of the four brass bands
need the between "requires" and "full"?
No, it doesn't.
Full orchestra is a technical term of music, and is generally used without an article. It does not have quite the same meaning as the ordinary noun phrase the/a full orchestra: it refers to something like a property or quality of a piece of music, rather than an actual physical orchestra.
A couple of example from the Corpus of Global Web English:
" 16 of her songs re-imagined and re-arranged with full orchestra."
"the Suite in G major for full orchestra,"
"explore the possibilities of instrumentation from solo piano to wind band and to full orchestra"
(Most of the examples of "full orchestra" in that Corpus do have an article).
In a case like this I would certainly not use "the", and I think most musicians wouldn't do so.
Edit: I've realised that this isn't a special property of the expression full orchestra, but of the way it's used by musicians. Many other expressions denoting musical forces behave the same way: choir, choir and organ, string quartet, full organ, wind band, brass band, small orchestra, piano solo, two guitars, piano four hands. All these can have works written/arranged/published for them or performed by them, never with an article.