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?
No it does not.
Once an attacker is declared attacking, it is attacking. Tapping or untapping it after it has been declared as an attacker will have no affect on that.
While untapping can (to my knowledge) never stop a creature attacking, you can use an effect like Tumble Magnet in a players "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. You cannot respond to a declared attacker by tapping it, which means you cannot wait until you know which creatures your opponent is attacking with to tap one down.
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 a permanent, it is still attacking. Similarly, if an effect only removes a permanent 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.