The spell description states:
When a creature enters the spell’s area for the first time on a turn or starts its turn there, that creature must make a Constitution saving throw. The creature takes 5d8 poison damage…
Compare this with the wording on Incendiary Cloud:
When the cloud appears, each creature in it must make a Dexterity saving throw… A creature must also make this saving throw when it enters the spell’s area for the first time on a turn or ends its turn there.
There is a difference of wording here, and the timing of the 2 spells is different (taking damage at the end of the turn on Incendiary Cloud opposed to the beginning of the turn for Cloudkill). RAW it would seem that the creatures in the area of effect will only take damage on the beginning of their next turn.
So, does this spell deal damage on the wizard's turn as it is cast? A good answer might not only consider rules-as-written, but also intent, comparison to previous editions, and relative power other spells. For example, Cloudkill is 5th level and Incendiary Cloud is 8th level.
Best Answer
It does not cause damage when first cast
According to Jeremy Crawford:
http://www.sageadvice.eu/2015/04/15/cloudkill/
Stinking Cloud (level 3) is worded similarly to Cloudkill, but has less powerful requirements. Notably missing is that entering the cloud does not apply damage, which could be because this spell doesn't do damage.