The output power of the transmitter is about +10dBm, and at this low power is unlikely to be damaged do to high VSWR. But never know..
The chip transmitter could be CYF115, or MICRF113 (marked OA9X9 in your circuit):
Check first if the LM1117-3.3 regulator have 3.3V at the output, and next check the crystal oscillator if is working. For these measurements you need a meter and an oscilloscope.
These modules are very cheap, and would be no big problem replacing them with newer ones.