Lifelines like unemployment insurance and federal stimulus checks were out of reach because undocumented people are ineligible for most government aid. Instead, they have relied on food pantries, lenient landlords and loans from friends. But after a sweeping move by lawmakers this week, New York will now offer one-time payments of up to $15,600 to undocumented immigrants who lost work during the pandemic. The effort — a $2.1 billion fund in the state budget — is by far the biggest of its kind in the country and a sign of the state’s shift toward policies championed by progressive Democrats. “I’ve met neighbors who have not been able to pay rent, or put food on the table, or been able to provide their children with a laptop when a public school has not been able to provide one,” said State Senator Jessica Ramos, a Democrat from Queens and lead proponent of the fund.