Those look like vapor canisters. The vapor canisters capture the fuel vapor as it expands in the fuel tank and absorbs it into some activated charcoal. At some point in time a solenoid opens and allows the engine to suck fresh air through the charcoal in the canister and pull the vapor into the engine to be burned.
The chewed canister will not cause the alternator problem. My guess though is whatever animal chewed on the canister also chewed on the alternator wiring. And for what it's worth, I'm betting that is the bottom of the alternator just above the canisters in the picture.
