A government agency whose mortgage guarantees help funnel billions of dollars to millions of home buyers ran out of money and authority yesterday, a situation that lenders estimate will prevent 10,000 families a day from getting federally backed home loans.

The Federal Housing Administration, caught under an avalanche of homeowners seeking to refinance mortgages as interest rates fall, reached its loan credit limit of $57.4 billion by early afternoon and told lenders to stop taking applications for mortgages insured by the agency.

But even if the FHA had not reached its credit limit, the agency would have been forced to issue the same order because its authority to guarantee loans was due to expire at midnight last night.

FHA's sudden inability to conduct business comes just three weeks after another key federal housing agency, the Government National Mortgage Association, better known as Ginnie Mae, hit its credit limit of $65.3 billion and had to stop issuing guarantees to owners of mortgage-backed securities.

Ginnie Mae is a key link between lenders and the markets that provide cash for home loans backed by the FHA and the Veterans Administration. It guarantees payment on FHA and VA-backed mortgages that have been bundled into mortgage-backed securities, which are bought and sold by investors.

FHA and VA loans, which account for 20 percent of all home loans in the United States, are especially important to first-time and moderate-income home buyers.

The Senate worked into the evening to pass an emergency measure extending the authority of FHA and Ginnie Mae. But the measure differs from a House bill passed last week, and working out a compromise could take weeks.

And even if Congress passes legislation in the next several days, it cannot take effect without President Reagan's signature, and he will not return to Washington from an economic summit in Japan until May 9.

The delay could cost consumers. Industry trade groups say it could add 30 days to the wait for obtaining financing and one-half percentage point to the price of borrowing money to buy a house.

The Mortgage Bankers Association of America said that FHA and VA loans now take about 60 days to process and are running about 9.5 percent.

Warren Lasko, executive vice president of the association -- a trade group that represents the largest group of home loan lenders outside of the savings and loan industry -- said the exhaustion of FHA's credit authority could drive up rates for government-backed loans by half a percentage point.

He said it also could push up interest rates for conventional mortgages, which now are running about 9 1/2 to 10 percent.

An official at the Department of Housing and Urban Development, which oversees the FHA, said, however, that FHA authority ran out for several days in March and "there was no real effect on rates going up."

Silvio DeBartolomeis, acting FHA commissioner, said the agency will insure any mortgages that already have been committed to prospective home buyers, but no new applications will be accepted.

The House last week passed an emergency measure to extend the credit authority of Ginnie Mae to $100 billion through Sept. 30, the end of the government's fiscal year, and to extend the credit authority of FHA to $95 billion.

The Senate version would raise Ginnie Mae's limit to $125 billion through Sept. 30. But it would grant FHA an additional $17 billion through June 6, when the Senate hopes to have a more comprehensive housing bill ready.