Does ward trigger when a spell or ability targets a permanent with ward that is already on the stack?

Ward triggers any time a spell or ability that targets the ward-bearing permanent is put onto the stack (CR 702.164b). The trigger condition is simply that a spell or ability targets the permanent — there is no exception carved out for spells or abilities that were already on the stack before ward was granted.

However, the most common scenario where this matters is when ward is gained after a spell or ability is already targeting the permanent. If a spell was placed on the stack before the permanent gained ward, ward does not trigger retroactively, because the targeting event already occurred (CR 702.164b, 603.6). Ward only cares about the moment of targeting, not the moment of resolution.

If the permanent already had ward when a new spell or ability was put on the stack targeting it, ward triggers normally and goes on the stack on top of that spell or ability. The controller of the spell or ability must pay the ward cost or the spell/ability is countered (CR 702.164b).

Concrete example: Your opponent's Raffine, Scheming Seer (which has Ward 3) is already on the battlefield. You cast Doom Blade targeting it. Ward triggers immediately when Doom Blade is put on the stack targeting Raffine. Your opponent's Ward 3 trigger goes on the stack above Doom Blade. Unless you pay 3 generic mana when the ward trigger resolves, Doom Blade is countered.

In summary: ward triggers at the moment of targeting. If the permanent had ward when it was targeted, ward triggers. If it gained ward after the spell was already on the stack, ward does not trigger for that existing spell.