FPGAs are never what you want if you want to 'just run a program'. They seem to be widely misunderstood as a silver bullet when really they're more of a duct tape solution to certain far corners of the problem space.
The people suggesting the ESP8266 are on the right track.
Devices running openWRT should be able to do what you described. This page is a good starting point: https://wiki.openwrt.org/toh/start
An example cheap device: https://wiki.openwrt.org/toh/unbranded/a5-v11
Another option: http://www.acmesystems.it/arietta (cute, tiny little device)
I got my information from www.cnx-software.com and https://archlinuxarm.org/platforms/
Good luck.
https://github.com/RinusW/WiPy/tree/master/AiCWebserver
MicroPython has also been ported to the ESP8266:
https://github.com/micropython/micropython/releases/tag/v1.8
https://hackaday.com/2016/05/10/new-part-day-a-beaglebone-on...
But, no wifi. Pair it with a ESP8266?
Maybe you can get away with just the ESP8266 and NodeMCU?
http://www.seeedstudio.com/depot/LinkIt-Smart-7688-p-2573.ht...