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, it is attacking. Tapping or untapping it after it has been declared as an attacker will have no effect on that.
506.4b Tapping or untapping a creature that’s already been declared as an attacker or blocker doesn’t remove it from combat and doesn’t prevent its combat damage.
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.
508.1f The active player taps the chosen creatures. Tapping a creature when it's declared as an attacker isn't a cost; attacking simply causes creatures to become tapped. [...]
508.1j Each chosen creature still controlled by the active player becomes an attacking creature. It remains an attacking creature until it's removed from combat or the combat phase ends, whichever comes first. See rule 506.4.
508.3. Third, the active player gets priority. Players may cast spells and activate abilities.
Besides the combat step ending, there are several ways a creature can be removed from combat.
506.4. A permanent is removed from combat if it leaves the battlefield, if its controller changes, if it phases out, if an effect specifically removes it from combat, if it's a planeswalker that's being attacked and stops being a planeswalker, or if it's an attacking or blocking creature that regenerates (see rule 701.12) or stops being a creature. A creature that's removed from combat stops being an attacking, blocking, blocked, and/or unblocked creature. A planeswalker that's removed from combat stops being attacked.
Your last chance to prevent a creature from attacking is during the Beginning of Combat step, before it can be declared as an attacker.
506.4a Once a creature has been declared as an attacking or blocking creature, spells or abilities that would have kept that creature from attacking or blocking don’t remove the creature from combat.