Would Chlorine or Chloramine gas show up on a VOC test?
Chlorine, no. Chloramine, maybe.

Monochloramine, $\ce{NH2Cl}$, is an inorganic compound, and not very volatile (when dissolved in water it forms more or less of an azeotrope, evaporating slightly faster than the water itself). So, not a VOC.

However, the way it does its job as a disinfectant is to basically shred organic compounds, like cell walls, key enzymes, etc, forming "organic chloramines" where one or more of the remaining hydrogens is replaced with an organic functional group. These now-organic compounds have a higher vapor pressure, so by design, chloramine is used to create VOCs. The idea, when used in tap water or pool/spa chlorination, is to allow the organic chloramines to boil off and disperse in the air, leaving clean water behind. That "chlorine smell" of a chlorinated pool is actually mostly organic chloramines evaporating out. In a more enclosed space like an interior room, these compounds can't disperse and so the smell builds.