Yes, if you research how to use an apple remote with a raspi, you will find that it is easy to get an IR receiver working with it. I use raspbmc on mine with an old apple remote and it works fine.

I guess the easiest way to get it done is to run Raspbian as the OS since it's pretty user friendly and start from there.

Just a thought - if you can come up with the circuitry for an IR transmitter, you can copy the signal transmitted by your TV remote, say #'s 1 to 0 and assign those to transponders 1-10. That way, at least you can test the lap counter software using your tv remote without having to have actual transponders running.

Once that's done, then work can be done on making the IR transmitters since you've established preset codes already.

Also, some sort of pre-registration of the allowed transponders should be in the software in case someone points a TV remote at your timer and comes out as 'unknown' in the lap timing results.