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."
Only fools and children dream of heroes.