Federal law does allow resident immigrants with a "green card" to own a firearm under certain circumstances. You will need to ascertain exactly what the State of Vermont requires, however. Find the authority in VT and get the requirements directly from them. Do not rely on folks on the internet as you may get more opinions and anecdotes than actual facts. I would start here: [Only registered and activated users can see links. ]
Welcome to the US and enjoy your stay!