By accessible, I both mean linguistically, and practically (could a merchant in Constantinople acquire a copy of the NT easily).
I imagine the answer's going to involve the fact that the Byzantime Empire is a 1000-year period; that's okay! I'm happy to have an answer that either focuses on the time period you're familiar with, or with an answer that discusses change over time.
Literate Christians continually read it in Koine Greek. This is for a number of reasons. Firstly, Greek develops over time, yes, but Byzantine Greek (throughout the period) doesn't change that much. Secondly, literature and literacy tends to be linguistically conservative in general - less susceptible to changes in the language which tend to emerge in speech. Thirdly, Byzantine Greek in its written forms tends strongly to a classicizing tendency - they studied ancient Greek and modeled their writing on the same, and thought of that as "Greek, as it is written", even if they didn't speak that way.
We have a sizable number of NT manuscripts that are Byzantine in origin. They didn't 'translate' the NT into Byzantine Greek (cf. there are some translations into Modern Greek).
I'm less certain on the practical level, but I can't see any reason a merchant couldn't pay to acquire a copy of the New Testament. NT texts were being copied and they weren't a restricted good.