You can then decide who might be the most appropriate person with whom to work.
However USDA's income limit in Schuyler County, NY is $74,050 for a 1-4 person family, so if you will have 4 people or less in your household your $77k/year is just over it. 5-8 people the household income can go up to $97,750 though so if you have 5+ people then your $77k/year would be OK. There are certain deductions you can take to reduce the amount of "eligible" income, such as $480 for each member of the family residing in the household who is under 18, disabled, or a full time student, child care expenses & disability expenses... so that may reduce your $77k/year of eligible income down to the $74,050 limit.
Then you have VA financing, which is 100% financing but only for veterans, and then there is SONYMA which is down payment assistance that when used in conjunction with a mortgage can come close to 100% financing for a $200k loan amount. http://www.nyhomes.org/Home/Buyers/SONYMA/DownPaymentAssista has that information on SONYMA's down payment help program.
Other low down payment options would be FHA's 3.5% down & 3% down with conventional financing.
Outside of VA financing, if you have enough deductions to get your eligible income to $74,050 OR you have 5+ people living in the home, then in my opinion USDA financing would be the best option for no down payment.
Even though I am in California I also do loans in New York, doing financing for someone buying a home in Broome County currently, if you have questions or need additional information I'd be happy to help.