It is certainly correct to write "neophyte writer".
A noun can be used to modify another noun in English
This is probably an example of the noun neophyte being used attributively. The attributive noun construction (also called things like "noun adjunct" or "compound noun" or "noun used as an adjective") is common in English. You may have heard of it already; if not, you should study it. In general, any noun can be placed before any other noun to modify it (the exact meaning of this construction is fairly vague).
A comparable example would be "master writer", which is acceptable even though "master" is not an adjective.
In addition, nouns can often be converted to adjectives
That said, it is fairly easy to convert a noun (or substantive) to an true adjective or vice versa in English. It would not be too surprising to me to see neophyte used as a true adjective, and in fact the Oxford English Dictionary classifies it as possibly being an adjective:
B. adj. (chiefly attrib.).
Recently converted, initiated, or ordained; inexperienced. Also: characteristic of a recent convert or of a beginner or novice; displaying inexperience.
The most adjective-like example listed in the OED in my opinion is
I think one factor that may faciliate this conversion is the etymology of the term: neophyte is borrowed from Ancient Greek, via Latin, and both of these languages generally have easy interchange of substantives and adjectives. In Latin, neophytus was perfectly acceptable as an adjective; it was borrowed from a Greek word meaning "newly planted".