I was supposed to have received a key from Neymar that would allow me to unlock the door to the chapel and get to the throne room. However, all that comes up is the lockpicking option. Is there a fix or am I going to have to redo the whole dungeon now?

If you mean the scriptorium key you get from Nedmar it opens a door on the same floor, in a room left from Nedmar's room.

Yeah. I found the door in question but it's not opening. Also, I can't seem to find the key in my inventory.

Oh. I guess it's a bug then :(
I thought you just mixed up the doors, like I did :(

SPOILER
Did you go down to the dungeon and free the guy has asked you to save? You will get the key after doing that quest.
The key will be located in the quest (crown icon)section of your inventory if you have it.