If someone declares an attacker and that creature does not have vigilance, if I untap that creature, for example with Burst of Energy, does it stop the creature from attacking?
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No, it does not.
Once an attacker is declared, it is attacking. Tapping or untapping it after it has been declared as an attacker will have no effect on that.
While untapping does not stop an attacking creature, you can use an effect like Tumble Magnet in a player's "Beginning of Combat" step to prevent a specific creature from being tapped to attack. It is important to note, however, that you must do this BEFORE your opponent taps their creatures to attack.
There are cards such as Gustcloak Savior which do behave in the manner described in the question. However, these cards explicitly remove the creature from combat. The untapping element is unrelated and incidental. If an effect only untaps an attacker, it is still attacking. Similarly, if an effect only removes a tapped attacker from combat, it is still tapped unless the effect explicitly untaps it.
No, once a creature is declared as an attacking creature, it remains so until removed from combat or when the combat phase ends.
The Combat Phase is broken up into five steps, in this order: Beginning of Combat Step, Declare Attackers Step, Declare Blockers Step, Combat Damage Step, End of Combat Step. During the Declare Attackers step, the Active Player follows rules 508.1a-508.1j, choosing attacking creatures and tapping them. After that, players have priority to cast spells and activate abilities.
Besides the combat step ending, there are several ways a creature can be removed from combat.
Your last chance to prevent a creature from attacking is during the Beginning of Combat step, before it can be declared as an attacker.