If I have a Archive Trap in the graveyard and cast a Snapcaster mage targeting the trap after my opponent searches their library would I be able to cast the trap for 0 or do I have to pay the full cost.

Thanks in advance,

Scott-

No, you may not pay 0. The 0 specifically replaces the mana cost, not "any cost to cast". Since a flashback cost is a separate type of alternate cost, it cannot be replaced by the "trap" cost this way.

Similarly to this, you cannot choose to Overload a card you've targeted with Snapcaster.