Problem driving 3V motors with TLC5940 and transistor

Vcc = 5V
Vmot = 3.7V

That data sheet is all about using a voltage higher than the maximum 18V. You are using just 3V7!

Remove the transistor all together. Replace the motor with an LED, and put a 2K2 resistor in place of the IREF resistor and see that your software works. Then change the IREF resistor to 200R and replace your motor. Note that this current is right on the limit of what the chip can supply, have you done the power calculations to make sure you don't fry the chip?