The power supply from a solar charger board has USB connection. But on my sensor board, I need only the VDD and GND. Can I simply cut one end of the USB cable and connect the VDD and GND wires to my sensor board?
Assuming your sensor board takes a 5V input, then yes.
If your solar charger board requires communications with the target USB device (to be charged) before it applies power then no.
Assuming you are taking less than 100mA then this will almost certainly work. However its not allowed according to the USB specification.
A fully compliant USB device should enumerate when it's connected. This is a process where the device tells the host (in this case charger) what it is and requests the current it needs. A device should take less than 100mA while this process takes place then may take more (up to 500mA) once its enumerated provided the host says its OK. If you fail to enumerate (which you will, since it's not a true USB device) you are supposed to disconnect and take no current.
Full details on the USB.org website
The charger would be within it's rights to disconnect you; however, I have yet to find any that actually do.