Hey! I gather from another post you’ve made here that you are in Florida! The good news is, according to the administrative code for the Fish and Wildlife Commission (68 A-6.003, Permit Requirement and Exceptions), you do NOT need a permit to raise or keep house finches - so you are not legally required to turn over your new charges to a rehabilitator (although you may choose to do so should rearing them prove too arduous).
This is of course assuming they are house finch eggs, which I am not an expert on identifying.
It sounds like you’ve already tried to contact your local rescues, but just in case, here is the list of licensed rehabilitators for the state:
If the ones closest to you are no longer in service, you may be able to contact someone further out who can give you specific advice over the phone.
As for rearing your babies, house finches are similar to other species of finch in their care requirements. Some folks have had success hand rearing zebra or gouldian finches. Google “handfeeding house finch” and you’ll find a couple guides that seem very thorough. I am not an expert as I’ve only had to handfeed small mammals and sick poultry, but my grandmother used to raise orphaned songbirds. Because they are very, very small when they’re first hatched, they require a lot of care and patience so you don’t accidentally overfeed or aspirate them.
I really like this guide, as it seems to cover all the important points very well, including setting up a brooder and sanitation:
http://www.finchinfo.com/breeding/handfeeding.php
Hand rearing birds this small is a mixed bag; you can do everything right and still lose them, so please be gentle with yourself.