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I'm only up to episode 10 of the anime, but it seems as though Nezuko has no source of sustenance at all, not even water. We know she slept for 2 years and that was said to provide an alternative to human food/blood. However, with the muzzle seemingly permanent and no food or water ever consumed in the show, is sleep her only source of energy?

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Demons

This are one of the abilities of a demon; (you can refer to the link to view the other abilities)

  • Immortality: Demons possess eternal youth and can live for centuries, sometimes even longer if they're incredibly strong. They can't die through conventional means, but can be killed by sunlight or with one of the Demon Slaying Corps' special Nichirin Blades.

  • Growing Power: The strength of a specific Demon depends almost entirely on how many humans they have eaten, and they can also grow stronger by receiving additional blood from Muzan, which is how the Twelve Demon Moons gained their overwhelming power. Some Demons, however, have a limit to how much they can use to grow stronger, and they might be not compatible with a higher amount of Muzan's blood.

Being that said demons don't need to eat humans at all to survive as they are immortal and they'd only do this because they wanted to become stronger so the answer is Yes, Nezuko takes the sleeping as form of nutrients to sustain herself though this not give her additional power because she's not consuming humans, she was just exhausted and use sleeping to make up for it.

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Yes, to gain energy and recover from injuries she sleeps and does not get any other form of nourishment. During the Mount Natagumo arc of the series when she gets hurt by one of the demons she passes out in order to recover(Episode 19).

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