Like many other photography applications, Aperture allows you to export your filenames according to preset naming patterns or to fully customize them. While the Aperture preset formats contains spaces, it's a very simple matter to remove them.
After you choose File > Export Version, look for the Export Name Format menu, and set it to Edit... . You will see the Naming Presets dialog. Notice that the presets contain spaces between the individual naming elements, like this:
If you don't want spaces, simply delete the spaces or replace them with underscores, hyphens, or other valid naming characters, like this:
Note that Aperture 1.1 or later offers a preconfigured export preset without spaces.