Can Riven, Broken Blade exhaust herself to pay Blade of the Exile's equip cost and attach it to herself? If so, does her exhaust trigger (spend buff to ready) resolve before the equip?

Yes — exhausting Riven is a legal way to pay Blade of the Exile's Equip cost, and yes her "when I become exhausted" trigger resolves before the Equip does.

Blade of the Exile reads "[Equip] — [A] or exhaust a friendly unit," so you pick her as the Equip's target, then pay by exhausting her (she must be ready — an already-exhausted unit can't be exhausted again as a cost, Rule 415.4). Because the exhaust happens during the process of activating the ability, anything that triggers off it gets placed on the chain above the Equip in a Pending state and resolves first (Rule 818.1.c.1). So you can pick "spend my buff to ready me" and be ready again before the Blade actually attaches — the cost was already paid, so readying her doesn't undo or fizzle the Equip, and she ends up equipped and untapped. That ready-me line requires she already has a buff; otherwise "Buff me" is the mode (and remember each mode can only be chosen once per turn). Note this all has to happen in your Main Phase during an Open State, not mid-Showdown (Rule 151.2).