There are enough points of access on Champlain that you should be able to find most species of fish without the need for a snowmobile. But if you want to bring one, they are allowed on the ice.
That being said, you should be aware that the Main Lake is entirely open water, and it seems unlikely that it's going to freeze at this point. So from Burlington you will need to go north to Malletts Bay or the Inland Sea, or south to near or below the Crown Point Bridge to find fishable ice.