In case the design has an integrated SPI Flash or I2C EEPROM, there are chances that CYUSB3014 can boot from the image files stored in either of them.
If not, then the device would need to be programmed via USB.
You may need to check the PMODE[2:0] of the CYUSB3014 in your design and confirm the mode it is set from the below:
Can you please elaborate on this statement, "Also, I'm having trouble sourcing this device. Is there a replacement?"