So, I'm making a aasimar celestial warlock and I was wondering if their abilities would stack together. I know that with the scourge and protector aasimar you gain extra radiant damage on your first hit with an attack or spell per turn, but the radiant soul ability for the warlock also increases the radiant or fire damage by your charisma modifier. Would these two abilities stack together or would that not be allowed. Looking forward to your opinions on the matter.....

They would stack just fine.  It's from 2 different sources.

The Protector/Scourge Aasimar will increase your damage by your lvl with radiant damage, but only to a single target and when you've got your Soul ability active.

The Celestial Warlock's 6th lvl will add your Charisma to all fire or radiant damages to a single target.

with Radiant Soul from Warlock remember it's against a single target.

It works with AoEs, but only against a single target and doesn't proc on Damage Over Time spells after casting.

It doesn't double proc on Flame Strike, you have to decide if it's going to trigger off of Radiant or Fire during casting.

If you cast a spell that does radiant or fire, they can stack, if you don't use a spell, the warlock bonus will not trigger.

Quote from Lunali >>

If you cast a spell that does radiant or fire, they can stack, if you don't use a spell, the warlock bonus will not trigger.

 This.