If I imagine really pie-in-the-sky style, I could replace most of the metals necessary in the actual launching of ships into space. There are carbon structures, silicates and plastics which could be used as pressure vessels and heat shields, etc. necessary to get a ship into space, especially if we assume the planet has a lower gravity and thus escape velocity than our Earth. Crystalline structures could be used for a sort of radar for navigation and radio for communication...
The problem I run across though is in the precision tools necessary for the discovery process and creation of all the systems necessary. Nothing melts, forms, keeps an edge, sharpens, measures, conducts, bends or holds its form like metal. Sure we have laser and water based cutting tools as well as diamond saws, etc., but it is hard to imagine a scenario where we discovered their usage without having metals.
Also energy storage and transmission becomes an issue. Electric eels and other animals that use true bio electrics rely on iron and copper to do so (in admittedly small amounts). To cut out all metals, you would have to use a sugar based energy storage system and bio-luminescence; nerves can only maintain a signal for short distances (less than 100 meters or so), so relays would be necessary every so often, and speed becomes a factor very quickly. No speed of light approximation here.
It becomes easier if you don't rule out all metals. Salts are fairly important to the development of life as we know it and by definition a salt is a combination of one metal and one non-metal. Also calcium, since you mention bone-based science, is a metal. If you only rule out iron, copper, zinc and nickel, you could imagine a civilization that was extremely different from ours. Perhaps these metals do exist, but in such small quantities that iron is considered more valuable than gold to us? Remember that just 200 years ago aluminum was the most precious metal due to its perceived scarcity. Your civilization could have gone totally bio-based for a long time before discovering a smelting process...
Anyway, I love Tyranids/Zerg, and would like to help you out more here, but if we base this entirely in science, this is really hard to do.