Hello! I'm from Bronxville and a lot of buildings here are co-op apartments from what I know of. There are a few owners of co-op units who rent their place but it needs to go under approval the board of directors of each co-op unit. They screen possible tenants I think. I lived here for about 5 yrs and it's wonderful. There are a few apartment buildings around here that are scattered that might be renting but your good bet might be to call a realtor in the area, check craigslist, or if you're around the area, look for the super of the building as they usually know..
If you are willing to go a little out of the main area of Bronxville, they have areas that they call P.O. Bronxville -- which means they are on the outskirts of Bronxville and uses that for their address -- although they are not really part of Bronxville per se. Prices could be lower there and still the areas are very good. If you take Metro-North, I would say the lowest(most south) you can go is between the Fleetwood station and Bronxville stations, going more north is good in general -- if you would like to be picky

.. More south of Fleetwood is more of a mix. I think it would help if you will know specifically which school/s you are considering and what's your budget. Bronxville itself is more of higher priced but if you got the money, go for it as it is very, very good. Don't even think to take an illegal u-turn as they give tickets very quickly
Goodluck!