For the migrant money distributed by FEMA (the billions is hyperbole) - the $650M is a budgeted item and so would have come from Congress, i.e. approved by a GOP led House and DEM led Senate, prior to Biden's signature.
The $650 million cited by Republicans refers to the Shelter and Services Program (SSP), which had a budget of that amount for Fiscal Year 2024, which ended on September 30. SSP allocates funds to migrant support groups, nonprofits, and state and local governments to distribute to those in need of shelter, food or medical care. While the funds are distributed by FEMA, they do not come from the agency's budget. The DHS Appropriations Act 2024 shows that the money is from the U.S. Customs and Border Protection (CBP) budget, which is then given to FEMA to hand out, "helping to facilitate the safe, orderly, and humane release of asylum seekers and families". CBP influences where the cash is spent, with a requirement to send operational data to FEMA on a quarterly basis. This data helps guide FEMA on where to allocate the funds, particularly to shelters that support CBP's border security mission.
In the latest spending bill, FEMA asked Congress for $33.1B and were given $20B. They have said they have enough for now and anyone in need should ask. They have also said that they could run out if there are more storms,
but Congress is in recess until after the election and can't provide more funds. I expect Biden will magic some up if needed.