PJ, you should contact the LAX coach for the open try out times. I suspect it is going to be during the summer BCT period. (Eric Seremet Head Coach 719-333-9298)
For clubs, you can contact the Clubs office to see if your sport has LOS. If you have LOS you go to your club practice instead of intermurals. However, if your club does not have LOS, you go to intermurals and your club on non-intermural days.
I was the USAFA Karate Team Captain from 96-98. At the time, that team had LOS and we trained 5 days a week for three hours. Way more than your intermural commitment. TKD folks were lumped into the club. There are multiple martial arts clubs at the Academy. When I was there, we had Ninjitsu, Judo, Traditional Karate, Competitive Karate, and Aikido. TKD was formed within the past 10 or so years.
LOS is quite coveted by clubs and not all are granted the status, so call to see if your club does have it.
You can find contact numbers for your club here:
http://www.usafa.edu/cadetFocus/cadetClubs/
Tae Kwon Do Point of Contact 719-333-8487
I believe TKD and Karate are both NCAA sports (certainly both Olympic) that USAFA does not participate in. This is a resources issue with how many mens teams vs. womens teams you have in your program, coaching staff, etc.
Best of luck!
P.S. We have a great history of TKD Competition at USAFA. Alex White was a team mate of mine and was a member of '01 (I think). He did quite well and finished 2nd at the Collegiate TKD Nationals in 2001.
http://www.ncta-usa.com/results/2001_NCTA_results.pdf I also think we was an alternate to the US National team that represented the US in the 2000 summer olympics in Sydney.