If you are playing as a custom faction and you design and save ships while you are playing that faction, your ships will reappear whenever you play that faction.
The only problem with this is that the custom factions must adopt a ship style that is based on one of the default majors. Therefore, any ship you design for your custom faction will also appear in the shipyards of the faction your ships are based on.
After all that I guess the answer is no. You can't make custom ships for your faction without modding.