UPDATE:
I contacted someone local who rescues sparrows, but when I went to check on the little guy, he had passed on. Thank you so much for the advice. I know a lot more about what to do next time, and I understand my mistakes.
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My dog brought in this baby, who is very clearly incredibly young. I believe he is a sparrow.
We located the nest, but it's in a part of our roof that's inaccessible to us. He made it overnight! So at sunrise, I made a small nest out of a cardboard box and set him near the entrance to his parents' nest.
He got very cold, so I put a Ziploc bag filled with heated water near him that I change out from time to time. I see both of his parents, but they are busy feeding what must be his siblings in the nest. They don't seem to want to care for him. They have come up to the box and looked at him, but they do not engage.
Because of this, I contacted wildlife rehab, but due to the bird flu, no birds are being accepted. There's nowhere near me in terms of rehab that I can get this baby bird to, so sadly, it seems to be mostly up to nature. I tried to feed him some wet dog food per Google since it's been many hours with no care, but he did not open his mouth, so I fear his time is coming.
Sometimes baby birds do not get to spread their wings, and it's really sad. Hopefully, a happy ending is on its way for this little guy. I am unable to take him in and get any of the required equipment. Any advice is appreciated!