Does Virginia code allow for splicing wires with "twist-ons" in a main breaker panel when connecting to a new side panel for purposes of installing a generator switch in the side panel?

Twist ons? Are you saying wire nuts which twist on? If you are extending some dedicated circuits to a transfer switch panel, wire nuts are about the only way you can do this. Do you have a local building dept. with an electrical inspector? He / she can verify this for you.

The instructions state"To bring the disconnected circuits in the main panel to the new side panel, the wires must be extended. Codes vary in this regard. Some, like ours, allow twist connectors in the main panel." Are the appropriate sized "twist connectors" allowed in the State of VA?