You don't choose a target for Atraxa's Skitterfang's last ability at the time it triggers. Rather, a second "reflexive" ability triggers when you remove an oil counter 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.

You choose which ability the target creature gets as the reflexive ability resolves. Players may not wait to see what ability you choose before deciding whether to respond to it.