They certainly had access to iron when Genesis was written, but in the time period it is meant to portray?
The story itself is clear that iron was available to Noah. The time discrepancy might pose a problem for dating the flood but not for construction of the ark.
It's easy to see why people writing in a world where iron is basically commonplace might think it existed in the time they envisioned the flood events occurred in.
Sure, that's plausible.
The problem is that we can only date the earliest known use of iron. The Bible itself is not solid evidence of earlier use but you can't use absence of evidence to contadict it either.
We can't be certain, but given what we know about iron manufacturing - it leaves behind evidence - the absence of the evidence is indicative of absence of iron manufacturing in that period.
Absence of evidence is the same argument that we use against the flood itself but absence of a worldwide flood is one thing and absence of small localized ironworks is another. People usually take it for granted that Noah started out in the Middle East but there's no particular Biblical reason to make that assumption. Tubalcain and his successors might well have been far from the popular archaeological venues, the "cradles of civilization". The use of iron isn't the least plausible part of the story.