YES! But you do need to be a credit-worthy buyer, you need to have proof of income, and you need to have access to money for a down payment, even if a small amount.
Income makes you a great candidate for an HDFC purchase of a sponsored unit listed by UHAB - the Urban Homesteading Assistance Board.
You need to know that HDFCs, are limited-equity coops designed for low- and median-income residents who plan to buy for the purpose of staying in an affordable unit in NYC, and while may be a good solution for many people here, it's not going to be the golden egg if you want a windfall by reselling in a couple of years. You will only get slightly more than you paid, but you get very affordable housing.
At UHAB we create, sustain and preserve affordable housing cooperatives.
We have plenty of coop apartment listings on our website at http://www.uhab.coop,
and we have a new buyer seminar every other week. Our sponsor units typically cost way below market, starting at $35,000. People need from $2,500 to $5,000 in down payment money.
But you could also buy in a privately listed Coop, and there are a variety of listings all over the city at our website. If you want the Bronx, we have several buildings, and I personally visited a nice one bedroom (eat-in kitchen, living room, bedroom, newly renovated entirely) in the Bronx that costs $68,000 (probably 20% down) on Fulton St. and E. 167th in Morrisania.
Note: The resale restrictions will limit how much you get back from the sale, so be sure the HDFCs are right for you.
Best of luck.