Can a stunned unit like Shadow still tap for its ability?

Yes, a stunned unit can still use an activated ability that requires exhausting as a cost.

According to the Core Rules (Rule 423.1), the Stun status only restricts a unit in two ways:

1. It prevents the unit from contributing its might to combat damage (Rule 423.1.b).
2. It creates a state where the unit cannot be stunned again (Rule 423.1.a.1).

Stun does not restrict a unit's ability to exhaust (e.g., to pay for an activated ability) or perform other actions. As long as the unit is not already exhausted, you may exhaust it to pay the cost of an activated ability.