Can a Wifi device be made to output a continuous 2.4 GHz wave?
Essentially any data radio's hardware is capable of doing this, as emitting a continuous carrier wave is one of the key parts of regulatory acceptance testing.

How you do it is not always a public feature, sometimes it requires an alternate firmware you have to obtain from the silicon manufacturer.

If you just need a low power carrier, the RF registers of the nRF24 series are very well documented and can be easily controlled from an MCU of choice with firmware you can quickly write from scratch.  This may be easier than trying to get a more complex wifi-type radio to do it.