Looking at the stat block for a keelboat, it says that it can take two actions on its turn from the list, the last being "fire ballista" and "move." Can the ship take the move action twice?

This would likely the the most common "two actions" -- move and move.

DxJxC has the right of it.

Been a while since I checked GoS. Here is the ship actions rules:

This part of the stat block specifies what the ship can do on its turn, using its special actions rather than the actions used by creatures. It even relies on its actions to move; it doesn’t have a move otherwise. The ship’s captain decides which actions to use. A given action can be chosen only once during a turn.

Looks like, it can't.