Does Ward trigger when a creature with ward is the target of an ability not just spells?

Ward is a triggered ability that watches for a specific event: a permanent with ward becoming the target of a spell or ability an opponent controls. This is explicitly covered in CR 702.20b, which states that ward triggers whenever the permanent becomes the target of a spell or ability an opponent controls while it's on the battlefield.

This means ward is not limited to spells. If an opponent activates an ability — such as tapping a Planeswalker's loyalty ability, using an Equipment's equip ability, or activating any other ability that targets your ward creature — ward will trigger just the same as it would for a spell.

When ward triggers, the opponent must pay the ward cost or the spell or ability is countered by the game rules (CR 702.20c). This applies uniformly regardless of whether the trigger came from a spell or an ability.

Concrete example: Your opponent controls a Shuko (Equipment) and activates its equip ability targeting your Grizzly Bears with Ward {2}. Ward triggers. Your opponent must pay {2} or the equip ability is countered and Shuko remains unattached.

Note that ward only triggers when an opponent is the source of the spell or ability. If you yourself target your own ward creature with a spell or ability, ward does not trigger (CR 702.20b).