I got invitation to visit a French school last week and have to be there in September. I am unable to get an appointment at the French consulate in NYC for the next two months. Can I request another French consulate to help and give an appointment as I definitely need it earlier?
Generally speaking, no, that's not allowed. The relevant Schengen regulation does leave the door open in special circumstances, but that's not how French consulates traditionally work. You have to apply to the consulate serving your place of residence and cannot simply choose one that's convenient for you, even in the same country.
However, it does not hurt to ask, either by trying to contact the NYC consulate directly or by lodging an application at another consulate, explaining why you do that (you can write a short letter to join to your application). Since it's a reasonable reason and it's out of your control, they might agree to see it as a valid justification to apply article 6(2) of the Schengen visa code but it's really up to them (see also Can I apply for Schengen visa from London as foreigner? and Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa?).
Some practical tips: Always be congenial and to-the-point in your communication, do not suggest you feel entitled to something or get angry – even if you have good reasons to! – but stand your ground and don't be obsequious either and you might get lucky. But it's a long shot.