The simplest way to power an off the shelf power adapter in this case is to connect another 120 to 24 VAC transformer 'backwards'.

Connect the 24 volt secondary of the transformer to your 24 volt AC source in the wall (at or near the thermostat location). The transformer primary will then provide full line voltage. Feed that 110-120VAC into your DC power adapter of choice.

You can wire the transformer 'output' to a standard AC wall outlet and then plug in your adapter. All this can be enclosed within your thermostat box, or tucked into the wall cavity behind.