By age 13, I would expect him to be riding a small frame bike with 700c tires. Don't expect it to be a forever bike, although you could probably size it slightly large and get a couple of years out of it.

I wouldn't plan on renting one for just races. That will get expensive quickly, and he really needs experience riding the bike, both for safety and training.

There should be lots of good used ones on the market. I wouldn't go out and immediately spend $800 on a bike. If he starts getting top tier finishes with an old steel bike, then you can consider something a bit better.

You can always sell the bike or pass it down to another deserving kid when he outgrows it. Get a good deal now and it should hold a lot of value.

Or, invest in upgrades like race tires/wheels that could be moved from bike to bike as he gets older.

Hmmm... looks like a lot of options from $100 to $500+

https://newyork.craigslist.org/brk/bik/5620862340.html
https://newyork.craigslist.org/fct/bik/5626759088.html
https://newyork.craigslist.org/fct/bik/5615360475.html
https://newyork.craigslist.org/fct/bik/5629104723.html
https://newyork.craigslist.org/brk/bik/5610824852.html
https://newyork.craigslist.org/mnh/bik/5628184788.html
https://newyork.craigslist.org/que/bik/5620215731.html
https://newyork.craigslist.org/brk/bik/5609599791.html
https://newyork.craigslist.org/que/bik/5585980724.html
https://newyork.craigslist.org/que/bik/5585634492.html
https://newyork.craigslist.org/jsy/bik/5621031179.html