The Oodnadatta Track doesn't have a dry or wet season, its slightly more likely to rain in January or February, but generally it could rain at any time of year, and if it does rain the track can be closed for days.    All you can do is watch the weather as you get near to it and if it has recently rained or rain is forecast make sure you get good local information on road conditions before starting out.   

What it does have is a HOT season, the average December temperature is 36.3 and over 40 is common.    http://www.weatherzone.com.au/climate/station.jsp?lt=site&lc=17043       Personally I don't think it would be terribly enjoyable in summer, but some people do do it.     We did it in April last year, the road was in reasonable condition at that time, some corrugations but not too bad if you drive to the conditions.   

We have a Pajero towing a 20ft van.    The road certainly doesn't require 4wd, at least not when it is dry.   Its a gravel road, fairly wide, and the main problem is the corrugations, though they weren't too bad when we were there.