You don't choose a target for Shire Shirriff's last ability at the time it triggers. Rather, a second "reflexive" ability triggers when you sacrifice a token this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

If Shire Shirriff leaves the battlefield before its last ability resolves, you can still sacrifice a token. However, when you do, the target creature won't be exiled when the "reflexive" triggered ability resolves. Similarly, if Shire Shirriff leaves the battlefield while the "reflexive" ability is still on the stack, the target creature won't be exiled.