Mirrors can be made out of just about anything: glass, metal, ceramic, polymer, wood, you name it. The mirror itself is just a reflective coating (typically gold, silver, or aluminum), and it can be as thin as a few nanometers. The trick is shaping it and convincing it to hold that shape, which is why we currently use rigid materials like glass and metal. Something inflatable might work, but more likely we'll just stick with lightweight metals like JWST's beryllium mirrors.

However, there's already a lot of rock up there. Asteroids are numerous and big. What's the concern adding more?