Can you sacrifice a permanent to pay a cost in response to a spell or ability?

When you activate an ability or cast a spell that requires sacrificing a permanent as part of its cost, you sacrifice that permanent as you pay the cost — this happens immediately as part of putting the ability or spell on the stack. You can do this in response to other spells or abilities, as long as you have priority and the ability itself has instant speed (or is an activated ability without a sorcery-speed restriction).

According to CR 602.2, activating an activated ability follows these steps: announce the ability, make choices, then pay costs — including any sacrifice required. The sacrifice is a cost payment, not an effect, so it cannot be responded to separately. Once costs are paid, the ability is on the stack and opponents may respond to it.

CR 118.12 confirms that a sacrifice is a cost that is paid all at once when the spell or ability is activated or cast. The permanent is gone from the battlefield immediately, before anyone can respond to the ability itself.

Important condition: the ability being activated must be usable at instant speed. Most activated abilities can be used any time you have priority (CR 116.1), but some specify 'activate only as a sorcery,' which would restrict when you can do this.

Example: Your opponent targets your Llanowar Elves with a Doom Blade. In response, you activate Viscera Seer by sacrificing the Llanowar Elves as the cost. The Elves leave the battlefield immediately as you pay the cost, so when Doom Blade resolves, it finds no legal target and is countered. Viscera Seer's ability is now on the stack and resolves normally.