I have a NodeMCU Esp8266 v3 (Lolin) and it works fine conecting Vin and G to my cell phone power supply rated 5.1V, 1000mA. This develop board has a aprotective diode (the closest to Vin), if you have a Multimeter o multitester you can check it measurent contininuity, if your diode is damaged the power applied to Vin never be enough to power up the nodemcu.
Regards from Mexico City.