Does it violate NEC to run an AC adapter's low-voltage output cable through a wall?
There's a better solution to this problem

Most low-voltage wall adapter outputs for low-powered ITE such as an IP camera qualify as Class II power circuits under the NEC, which means that Article 725 applies to the situation you propose.  However, the wire used for wall adapter output cables is not legal to stuff into the wall in the way you are proposing, as it's not a listed UL/NEC type CL2 or CL3 cable as required by NEC 725.135(A) and (M) (instead, it's some AWM type, at least on the adapter I had laying around handy).

So, you have two options at this point:

Run a CL2 cable through the wall

The perhaps-simplest way to do this would be to run a listed CL2 or CM cable through the wall -- two-conductor thermostat or in-wall speaker wire works, although you'll need to get jacketed cable instead of the normal unjacketed-pair stuff in order to comply fully with 725.139(E).  At each end point, you'd terminate it to a dual binding post or dual banana jack wallplate (you can get modular wallplates that can be set up this way -- look at the Leviton QuickPort line for one of many example).  You'd then chop the AC adapter's output cable in two, fit banana plugs to the chopped/stripped ends, and plug them in in the appropriate places, minding polarity of course.

Just make sure that the low voltage plate is on a separate bracket or box from the 120VAC, or that a listed divider separates the 120VAC gang from the low voltage gang.

Do it the old fashioned way

The "old fashioned way" is to use a proper Chapter 3 wiring method such as NM (Romex) cable (note, NOT flexible cord, or else it's as many lashes from an extension cord as it takes to acknowledge that NEC 400.8 is true, wise, and holy!) to run 120V power from a nearby circuit to a new receptacle located near the camera's home, which then can have the AC adapter run to it.

You'll of course want to turn off and figure out the circuit you plan to use for this, and use proper splicing and cable securement techniques.  Also, the power cable can't share a junction box with the Ethernet cable unless you shove a listed divider between two separate gangs, one for the Ethernet side and the other for the 120VAC side.