Use of a single cell lipo for VIN enough to power motors and board
So yeah, providing the board with 3.7V on the VIN pin will result in a massive under voltage on the supply rails due to regulator dropout. 

The LD1117 used on the input has a dropout of around 1.2V which is then followed by a diode which will have around 0.5V dropout - as a result the 5V rail will end up being 2.5V or so assuming a fully charged LiPo (4.2V - 1.2 - 0.5) or closer to 2V for a nominal charge (3.7 - 1.2 - 0.5).

The 3.3V rail will then be closer to 1.8V based on the second regulator which is only around 0.2V dropout. I'm honestly surprised the MCU actually runs correctly at that voltage, though it is probably just about within spec.



Bypassing the 5V regulator would allow the 3.3V rail to correctly regulate as it has sufficiently low dropout voltage to do so - you need at least 3.5V in order to maintain the output voltage of that regulator. As the LiPo discharges it will get down towards 3V, by which point the 3.3V rail will have begun to drop off again. So it is still not ideal, but should be workable for a large portion of the battery charge.

To bypass you would need to connect the 3.7V to JP5 pins 2 and 3 leaving pin 1 disconnected. This should ensure power is provided to all parts of the board which need it, and also mean that if you connect the USB port it doesn't start to charge the battery incorrectly.

You will still not have a 5V rail, however all parts of the board use 3.3V rails which are regulated using ultra low dropout regulators from the 5V line which should be able to maintain regulation even with a LiPo on their input.