Building a radio isn't anything like a practical project these days, they are just so cheap to buy - which is why it's difficult to find parts to build them with.
If you really want to build a radio (as a learning project, rather than you want a radio), I would suggest building it in a modular fashion - with the front end, IF, and audio stages all on seperate boards. I would also suggest probably starting by NOT using IC's and building a 'proper' IF amplifier with transformers and transistors.
But the best source for parts is really still an old radio!.