When I set up Eclipse do players have access to all the ship part tiles throughout the game? And is it acceptable for me to layout a couple of each tile on the Supply Board and top them up as players take them?
The supply board contains two panels for Ship Parts; those five in the left-most panel are available to everyone at all times. The other twelve in the right panel — distinguished by the Technology icon printed on the upper right corner of the tile — are only available to users who have researched the corresponding technology.
Unlike most other pieces, Ship Parts cannot be exhausted and as such are always available. Even if the physical tiles themselves are somehow depleted, replacements may be used as needed. Since the actual quantity of tiles is irrelevant, there's no reason they can't just be drawn/topped-off as they're acquired rather than all placed on the Supply Board during setup.