ESP8266 modules are tiny, have a 32-bit processor with 8 pins and wifi.
How big a battery to run how long? Not very big = not very long.
A servo needs to be able to move to and hold a position. Those legs don't. The driver can be simpler.
Feet that slip forward but hold to push when moved back..... or move in ellipses higher on the swing forward. It might be possible to drive each side with 1 motor and some mechanics. Watch Ralph Steiner's old short feature movie on youtube for inspiration.