I was trying to make a character sheet for an NPC but I’ve ran into an issue with the backgrounds. I was wanting him to have the Mercenary Veteran Background from the Sword Coast Adventurers Guide but none of the backgrounds from the book are available for selection. I don’t own the book but it’s been shared with me through a campaign. I’m not sure if this is an issue with other books but I checked on both the app and the website and the issue remains.

That content isn’t being shared “with you,” it’s being shared with that campaign itself. You only have access to it in that specific campaign. You will have to build your character in that campaign to use the content being shared with you there.