Is 'request denial' a phrase?

It is not a "phrase" in the sense of a fixed phrase - it is a simple collocation.

"Request denial" is two nouns. This is a common and productive way of producing compound nouns.

Nouns can act adjectivally to modify the following noun in this way:

Noun1 noun2 = the noun2 associated with noun1.

The Language Department = the department associated with language.

A beer bottle = a bottle associated with beer.

There can be several nouns together:

A beer bottle shape = a shape associated with a bottle that is itself associated with beer.

NB: When nouns are used together like this, they form one compound noun and cannot be separated.

The modern {Language Department}

• The Language modern Department

Because noun1 acts adjectivally, it is not common for noun1 to be plural.

But The modern Department of Languages

The Department of Modern Languages

-> you will note that these mean different things.