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Esp8266 based wifi switches can be had quite cheap (sonoff is one of the brands)

looks perfect for my needs, is there any REST API documentation?

Dunno, but you can flash any esp8266 firmware onto it and roll your own API (or other smarts)

I mean write my own script to turn on/off the switch such as using wget or libcurl

Yeah, you could flash a simple Arduino sketch with such an API, and use that, assuming you can't access the in built API