As a member of the Empire, you can't build anything. However, you could own a factory that was managed by a ship-production faction, had a ship datacard assigned, and have production started on a ship by a member of that ship-production faction (who is assigned as operator of the factory). Then the ship would be produced in your factory and you would be the owner. As long as the datacard is assigned the facility(/station) then it doesn't matter who owns the factory/station, nor which faction (of the correct type) is managing and has their member operating.
Maybe not too clear, but I couldn't think of a better way to say it.
"May the Grace of Ara go with you, and His Vengeance be wrought upon your enemies."
Here's the site with all of the production guidelines read more about it here.
You need to have the DC, the materials, worker NPCs and have enough money to pay for production cost. You tool the DC to the factory, sort of like assigning a blueprint. Then after that's done which takes a couple hours, you can go to produce and start making it. You also need to be a member of the respective thing you are building. For EX if your building a ship you need to be in a ship production faction and have the necessary privs to build.