Can you cast a spell that targets a permanent in response to that permanent being sacrificed?

Sacrificing a permanent is most commonly a cost paid to activate an ability or as part of resolving an effect. Costs are paid during the process of putting an ability or spell on the stack, before any player receives priority to respond (CR 116.2, 601.2h). Because the permanent leaves the battlefield the moment the cost is paid, it is gone before anyone can cast a new spell targeting it.

When you cast a spell, you must choose targets at the time of casting, and all targets must be legal at that moment (CR 601.2c). A permanent that has already been sacrificed and moved to the graveyard is no longer on the battlefield, so it cannot be a legal target for a spell or ability that targets "a permanent," "a creature," "an artifact," etc. Attempting to target it would be impossible from the outset.

It is worth distinguishing this from a triggered ability that says "sacrifice a creature" on resolution: even then, the sacrifice happens during resolution, after players last had priority, so there is still no window to target the permanent before it is sacrificed.

Concrete example: Your opponent activates Goblin Bombardment by sacrificing their Llanowar Elves (paying the sacrifice as the cost). You cannot respond by casting a spell that targets Llanowar Elves — it left the battlefield the instant the cost was paid, before you ever received priority.

The only partial exception is if a spell or ability with no targets is on the stack and a separate effect would later cause a sacrifice on resolution — but even then you cannot pre-target the permanent to save it, because spells resolve sequentially and priority windows occur between spells/abilities, not in the middle of one resolving.