What keeps working, and what stops
Wood stoves and wood inserts burn and radiate with no power at all. Their blowers stop, which reduces how far the heat travels from the appliance. Pellet stoves and pellet inserts stop entirely and some push a small amount of smoke into the room as the exhaust fan winds down. Gas appliances vary: many need power for ignition and controls, and some models are designed for standing-pilot operation and will run through an outage. Which yours is, is a specification question worth settling before you need the answer.
Why this matters more here than most places
Rural Vermont loses power in winter storms, sometimes for days, and the same weather that causes the outage is the weather that makes heat non-negotiable. The strongest practical argument for keeping a wood appliance in a rural house even where it is not the primary heating system. It is the one that needs nothing but fuel and a working flue, and it is why many households here have one alongside something more convenient.
Making a pellet appliance survive an outage
A solvable problem if you plan for it. A battery backup sized for the appliance will run it for some hours; a generator will run it indefinitely. Either needs to be set up before the outage, not during one. What is not solvable is the appliance itself. It is a machine and it needs current. Anybody relying on pellet heat as their only heat in a rural Vermont winter should have one of those two arrangements or a second appliance that needs no power.
The overlooked risk: the appliance you cannot use
The situation that catches people is a wood appliance that is perfectly capable of heating the house but has not been serviced, or whose flue has not been inspected, and which therefore cannot safely be relied upon when the power goes. An appliance you are depending on in an emergency should be the one you maintain most carefully instead of the one you neglect because it is the backup. The argument for servicing it before the season regardless of how little you expect to use it.