A lot of engineering companies are completely private, and have no allegiance to any one government or organization, so in that way I suppose there would be. It’s a bit hard to compare them to SpaceX, because all the technology already exists and it isn’t really a thing to have governments directly involved in the design and production of spacecraft (the exception being Top Secret Death Weapons such as the Death Star, for which the Empire did have its own scientists and engineers, but the actual construction and the bits of technology that already existed would have been carried out by a private corporation under contract). A notable example would be Santhe/Sienar Technologies, who supplied military vessels to the Republic, the Trade Federation, the Empire, the New Republic (Legends) and the First Order. They would be the closest to a SpaceX sort of thing I can think of. I don’t think anyone would really go through financial effort to keep these companies from selling to rivals, it’s understood that they don’t usually have any one exclusive affiliation (though the New Republic did introduce arms restrictions that limited their abilities to sell to the First Order).
Hope that helps!
~ Jacen