Attaching an Equipment with its enters-the-battlefield triggered ability isn't the same as using its equip ability. You don't pay mana for the attachment, and the timing restrictions for equip abilities don't apply.

Mithril Coat doesn't enter the battlefield attached to a creature. Instead, the Equipment enters the battlefield and then a triggered ability attaches it to a creature. You may cast Mithril Coat even if you don't control any creatures.

If the target creature becomes an illegal target, the Equipment remains on the battlefield unattached.