It works the other way around: you use the wet bulb temperature to find relative humidity. I can't think of any direct use of wet bulb temperature.
The classic way to measure wet bulb temperature, is, well, using a wet bulb.
The reservoir of a thermometer is kept wet by covering it with a wet wick. The water will evaporate and cause a temperature drop. The difference between dry bulb and wet bulb temperature is an indication for the air's humidity.
Electronic humidity sensors are mainly capacitive or resistive. Sensirion is a manufacturer of small sensors which combine temperature measurement with relative humidity measurement. They communicate with a microcontroller via I2C. Sensor size is just 3mm \$\times\$ 3mm.
While Sensirion mentions Conrad, Farnell, Newark and RS as distributors, availability may be a attention point.
Introduction to humidity
Choosing a Humidity Sensor: A Review of Three Technologies