According to Scripture, is sacrifice a requirement for salvation? Regarding salvation, I was always under the impression a sacrifice had to be made (and as Christians we know Jesus was the ultimate sacrifice for all sins), but looking through the Bible I see instances of God forgiving sins without requiring a sacrifice..

2 Chronicles 7:14 God forgave without a sacrifice. In Matthew 9:5 Jesus forgave without a sacrifice.

I'm used to hearing the judge analogy which states "A good judge can't dismiss sin, there has to be a punishment". According to Scripture, if God is righteous/a good judge, how can he forgive sins without a punishment/sacrifice?
What is salvation according to Scripture?

The OP has the right understanding. People are saved from their sins by Jesus


  Matt 1:21 (RSVCE) 21 she will bear a son, and you shall call
  his name Jesus, for he will save his people from their sins.”


Why is a sacrifice required?

Again OP has a right understanding. God is good, merciful and just. In justice, atonement for sin needs to be made, in his mercy, he has provided for forgiveness of sin. He has required sacrifice to achieve both ends. Saint Paul explains in Roman 5:1-10


  5 For every high priest chosen from among men is appointed to act on behalf of men in relation to God, to offer gifts and sacrifices for
  sins. 2 He can deal gently with the ignorant and wayward, since he
  himself is beset with weakness. 3 Because of this he is bound to offer
  sacrifice for his own sins as well as for those of the people. 4 And
  one does not take the honor upon himself, but he is called by God,
  just as Aaron was.
  
  5 So also Christ did not exalt himself to be made a high priest, but
  was appointed by him who said to him,
  
  “Thou art my Son, today I have begotten thee”;[a] 6 as he
  says also in another place,
  
  “Thou art a priest for ever, after the order of Melchiz′edek.” 7 In
  the days of his flesh, Jesus[b] offered up prayers and
  supplications, with loud cries and tears, to him who was able to save
  him from death, and he was heard for his godly fear. 8 Although he was
  a Son, he learned obedience through what he suffered; 9 and being made
  perfect he became the source of eternal salvation to all who obey him,
  10 being designated by God a high priest after the order of
  Melchiz′edek.
  
  a. If Jesus was to be mediator, he had to have a human nature
  like ours, and, moreover, he could not appoint himself, but had to be
  appointed by God.
  b. Greek he


What is sacrifice?


  SACRIFICE The highest form of adoration, in which a duly authorized
  priest in the name of the people offers a victim in acknowledgment of
  God's supreme dominion and of total human dependence on God. The
  victim is at least partially removed from human use and to that extent
  more or less destroyed as an act of submission to the divine majesty.
  Thus a sacrifice is not only an oblation. Where an oblation offers
  something to God, a sacrifice immolates or gives up what is offered.
  In sacrifice the gift offered is something precious completely
  surrendered by the one making the sacrifice as a token of humble
  recognition of God's sovereignty. (Etym. Latin sacrum, holy, sacred
  + facere, to make, do.) - Dictionary: SACRIFICE | Catholic Culture


All the acceptable sacrifices in the Old testament were a type of Jesus' sacrifice. Jesus' sacrifice is the one that actually effects forgiveness of sins past, present and future.


  Matt (RSVCE) 28 for this is my blood of the [new] covenant,
  which is poured out for many for the forgiveness of sins.