I have a paladin in my campaign that through "Oath of the Ancients" has the Speak with Animals spell, but that's a ritual spell and Paladins don't have ritual spell casting as a class feature. Can he still cast the spell or does he have to get something first?
The ritual tag identifies spells that can be cast as rituals, not spells that must be cast as rituals. Your Paladin can cast Speak With Animals as a regular Paladin spell using the normal rules.
From the SRD:
Certain spells have a special tag: ritual. Such a spell can be cast following the normal rules for spellcasting, or the spell can be cast as a ritual.