Measuring current drops voltage?
This is the "burden voltage" of an ammeter - when a 'shunt resistor' is used to measure the voltage drop due to current flow, then the voltage seen by the circuit being powered is lower.  Multimeters, because of their wide sensing range, tend to have relatively high shunt resistors, which causes the circuit being powered to see significantly less voltage, and so some things don't work right.

I'd recommend putting a very low value resistor in series with the 3.3V rail that's powering the wifi module (say, 100mohm), and then measure the voltage across that resistor &amp; do the Ohm's Law calculation to work out the current.