Many states do not cover Green Card holders under the Federal Govt funded 'Medicaid/ Medicare' programs.
But almost all states have their own programs which help individuals and especially for kids who do not get coverage under above programs.
Usually, under such kids' health insurance programs Green Card holding kids are covered.
For NY, your kids will most certainly will get coverage according to NY health coverage application instructions.
SECTION D Citizenship
This information is needed only for those people applying for health
insurance. Pregnant women do not have to complete this section. To
be eligible for health insurance, other persons age 19 and over must
be citizens or must fall within one of many immigration categories.
Children who are New York State residents and who do not have
other health insurance are eligible, regardless of their immigration
status.
PUBLIC CHARGE INFORMATION
The United States Citizenship and Immigration Services (USCIS) has
said that enrollment in Child Health Plus,Medicaid, PCAP or Family
Health Plus CANNOT affect a person’s ability to get a green card,
become a citizen, sponsor a family member, or travel in and out of
the country (except ifMedicaid pays for long-term care in a place
like a nursing home or psychiatric hospital).
The State will not report any information on this application to
the USCIS.
For more info check;
http://www.health.state.ny.us/nysdoh/chplus/
http://www.ins.state.ny.us/website2/hny/english/hnyeci.htm