I am trying to use
L293D with relay to power a 12V DC motor. Can I use arduino's 5V to provide L293D inputs 4 and 8 such as in the drawing? Or do I need to use a separate source, e.g. a power module?
it looks like that motor controller chip has separate logic level and motor power inputs.
VCC1, at pin 16, is a low current logic level power input. You should be able to drive that from the Arduino's power supply.
VCC2, at pin 8 is the motor power input. You should NOT drive that from your Arduino's power supply. You should have a separate power supply for that (+5V with enough current for all the motors you are powering - 600mA is the max that chip can drive for a single motor, so a 1A power supply for VCC2 would be fine.
It looks like you are feeding the Arduino's +5V line into VCC2. DON'T DO THAT.