Can I gather all the items we need without buying the $200 kit?
Yes, indeed. Even tho I've only competed in Forensics C and never in Crime Busters B, the materials acquisition principles should be similar. Many of the qualitative analysis substances can either be sourced from your school's chemistry department or from the general store. Polymers are a bit trickier, but for me, my local JoAnn's Fabrics store let me have 1-inch samples of all the fabrics that were 100% of the fibers I needed (like 1 inch wide, as long as the length of the fabric on the roll), for hairs, you can collect human hair samples from yourself or your friends (haircuts!), and you can collect dog and cat hair from your or your friends' pets. The plastics are a bit trickier, especially since many sources of plastic that are around you may not necessarily be pure samples of the polymers they're made of. And then all the reagents and lab equipment should also be obtainable from your school's chemistry department, although I don't know much about your particular school so I'm not necessarily qualified to comment on feasibility of obtaining materials in that sense.
Hope this helps!