Thank you for the updated wiring diagram and pictures. I noticed your system keeps getting more complicated with each post. I recommend simplifying everything until you find out what is causing a problem. Additionally, in your latest wiring digram, you show a resistor between VOUT and GND. I do not see any reason for it; it is likely just causing unnecessary current draw. You should not need to connect anything to the VOUT pin on the logic side of the board. Also, you should not be connecting the M#CS pins to ground. Those pins are outputs, so they should either be connected to Arduino pins defined as inputs or left disconnected. Another thing I noticed is that your power supply is set to 5.6V, which is very close to the 5.5V minimum operating voltage of the VNH5019 shield.
Could you try increasing the voltage (maybe to around 7V) and removing all of your connections to the logic side of the board except for the required ones (shown in color) in the diagram I posted in my first post (GND, VDD, M1INA, M1INB, M1PWM, M2INA, M2INB, and M2PWM)? The M#EN pins are pulled-up on the board, so they should not need to be pulled high externally.