All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If it is proven that someone has committed theft based on either evidence or confession, then the authority has to implement the prescribed corporal punishment, which is cutting off the hand. Allah Almighty says (what means): {[As for] the thief, the male and the female, amputate their hands in recompense for what they committed as a deterrent [punishment] from Allah. And Allah is Exalted in Might and Wise.} [Quran 5: 38]
Muslims as individuals, however, are not entitled to kill the thief or cause him impairment, as this will result in a security breakdown and the spread of chaos.
The fact that the authorities do not implement law procedures concerning the thief and that they might release him the same day is a lesser evil than the chaos that would spread among the people when every individual was left to implement the corporal punishment or punish the offender by himself.
If you ever face the need to prevent a thief from stealing and he resists and you do not find another alternative way to stop him except by killing him or causing him some sort of harm, then you are blameless for this, as long as Allah, the Exalted, knows that you were just defending yourself. However, if you do this, then you will need to prove his crime and assault against you before the authorities, otherwise you will be deemed guilty.
Allah knows best.