Agonizing blast does not show up on a non warlock character on the eldritch adept feat. Even though the character has picked up eldritch blast as an extra spell via spell sniper.

Only prerequisite for agonizing blast is that the character has eldritch blast.

From the text of the feat

If the invocation has a prerequisite of any kind, you can choose that invocation only if you’re a warlock who meets the prerequisite.

Agonizing blast has a prerequisite, thus you cannot select it if you're not a warlock

Studying occult lore, you learn one Eldritch Invocation option of your choice from the warlock class. Your spellcasting ability for the invocation is Intelligence, Wisdom, or Charisma (choose when you select this feat). If the invocation has a prerequisite of any kind, you can choose that invocation only if you’re a warlock who meets the prerequisite.

Whenever you gain a level, you can replace the invocation with another one from the warlock class.

(Bold and underline mine)

(edit: um, jinx? :D    )

Yes I know it's a warlock invocation. But eldritch adept allows for non warlocks to pick up warlock invocations.

see Eldritch adept.

However, agonizing blast does not have a prerequisite other than that you have the eldritch blast cantrip.

Agonizing Blast

 Prerequisite: eldritch blast cantrip

When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit.

Seems kind of silly that you have to a warlock to pick up agonizing blast.

If your DM agrees that it's a silly restriction, then you can work something out with homebrew. But D&DB is only going to support the rules as written.

The feat adds its own prerequisite of also being a Warlock.

Yeah I saw that in bonnkansans reply.