I have a Pi Zero W, which works when connecting to wireless when powered via the 5v USB and also has a decent console resolution on the HDMI port.
However, when powered with 3.3v using a custom carrier board both the console resolution is lowered and the zero reboots as soon as it is told to connect to the wireless network. When this happens, leaving the wpa_supplicant.conf file intact will cause an infinite reboot loop which only stops when the wpa_supplicant.conf file has the network section removed using a different machine.
I am at a loss.. does the wireless chip require 5v to operate?