If the creature becomes an illegal target by the time Chandra's Ignition tries to resolve (perhaps because another player controls it or it's left the battlefield), Chandra's Ignition won't resolve and none of its effects will happen. No damage will be dealt.

Use the power of the target creature as Chandra's Ignition resolves to determine how much damage it deals to each other creature and each opponent.

The creature is the source of the damage, not Chandra's Ignition. For example, Chandra's Ignition can have a white creature deal damage to a creature with protection from red.