The only way that I know in how to move a planet is to have the action "Wormhole" set upon it. This is something that is actually plausible for Star Control's Arilou race. You can see all possible actions in the ModifierEffectTypes.xsd file.
How would shipyards work in conjunction with such if possible?
Well, when a shipyard is un-anchored and moved, then construction is stopped for that turn. So a shipyard would not be able to move at the same pace. In case of previously mentioned Arilou, I have removed shipyard decay entirely, which is also plausible lore wise as they could be able to teleport resources to the shipyard.