ESP8266 operates at 3.3V and consumes around 100mAmp current, for designing of its power supply you can use LM1117-3.3V Low drop out (LDO) linear regulator which is having current capacity of 800mAmp. Remember that, this regulator is low drop out regulator. It is better to give 5V as input to the regulator to avoid over heating of regulator. You can create small heat sink on PCB using copper area.
At lower input voltage below 4.7V device will get programmed but it will not run your code.