Yes, both government agencies and higher education institutions (universities) can and do sponsor H1 and green card. All they need to be convinced is that no citizen who applied for that position is as qualified as you the foreign national. I just heard that a friend of mine got sponsored for green card and appointment as assistant professor in a good university even before his Ph.D. viva. Good for him. When I was in Florida, I had a floormate appointed by the county government and got his green card.