First you need to get an advanced c-card, like CMAS** or PADI AOW.
No, you don't.
I have an NACD cave card with no AOW. I know one of my dive buddies has IANTD Cave without AOW as well. I took nss-cds basic cave (before nacd had intro with doubles) without AOW.
The general attitude around cave country seems to be that AOW cards are "purchased" and they really don't hold much value. If you can find a quality instructor, I'm sure the skills would better prepare you for cave training, but it's not a prereq, I think NAUI might require it?
Either way, the basic things that are different between the various training levels are
Cavern- light zone only, typically 130ft from the surface, penetration+depth.
Basic/Intro - Typically 1/6th of doubles (or 1/3 of singles, if single tank cave diving is your thing lol).
Cave 1 (GUE)- basically intro, but they calculate 1/3 of 2/3 for penetration gas.
Cave 1 (NAUI) - 1/3rds and 2 navigational decisions
If I had it to do over again, I'd take cavern and intro together, and then later on take apprentice and cave together (or cave 1 then cave 2 if done that route). Looking back, I think GUE and NAUI have it right-- take an intro to tech class, and THEN do your first cave class, skipping cavern. I think there's so much overlap between cavern and your first cave class that the flow of information would have been easier to manage had I combined them.
The NAUI program is interesting because it allows a new cave diver to go a bit further, and provides the training to do so. It eliminates the risk of breaking 1/6ths or going down one of the common jump passages that most intro/basic divers feel after a few dives, but limits the decisions to 2, so they can't get extremely far off mainline.