The part of a modern firearm that has the same function as the lock on a muzzleloader is the action.

What the lock does

On a traditional muzzleloader, the lock is the whole firing mechanism that:

  • Cocks and releases the hammer.
  • Strikes a primer (flint, percussion cap, etc.).
  • Ignites the main powder charge so the gun fires.

Modern equivalent: the action

On a modern firearm, the action is the assembly that:

  • Loads and chambers the cartridge.
  • Cocks and releases the firing pin or striker.
  • Ignites the primer and allows the shot to fire, then ejects the spent case.

Because both the lock and the action are the mechanisms that actually make the gun fire (rather than just controlling or guiding the shot), the action is considered the direct counterpart to the muzzleloader’s lock.

TL;DR:
If you see the question “Which part of a modern firearm has the same function as the lock on a muzzleloader?” the correct answer is: the action.

Information gathered from public forums or data available on the internet and portrayed here.