The General Gameplan

This matchup is very dynamic since Zappa's various summons play differently and give him different tools. You're going to be able to run away against the dog, but you have to go in against the sword. We'll have to cover a lot of alternate normals and gameplans, so expect some sections to be quadrupled-up. The general gist of the plan though is something like this:

Zappa Naked: Zappa will try to use the centipedes summon attack since it's a DP without the input (IIRC it's a 236 instead of a 623). Bait summons and punish where possible since it's unsafe on block, but be willing and ready to go in. Unlike the fight against ABA, it's easier for Zappa to get positioning to summon safely, and the reward for him getting a summon isn't as scary as the reward for ABA landing the right counterhit into keygrab or chugging a bloodpack.

Dog: Try to outrun the dog and generally keep your notes on the ground. This will either hit the dog or force Zappa into the air where it's easier to deal with him. If Zappa is up close or has you sandwiched, learn how to countermash his gaps, dead angle or blue burst to end his pressure. You don't want to let him get going.

Sword: If the sword is between you and Zappa, be cautious with your movement and don't commit too much when you swing at him or you can let him get started. He's likely to push you into the corner or out to full screen if he hits you or gets you blocking, but if you can get past the sword, you have an opportunity to hit Zappa to get rid of the sword. Watch for his DP and his jumping spin attack which move the sword to Zappa's position (and his jumping spin attack also helps him move long distances quickly to escape. If you've gotten past the sword the risk/reward is better for committing/swinging, but if you're too predictable Zappa will use those two tools to reset the situation and maintain control.

Ghosts: Zappa can rather safely keep his distance and toss ghosts to try to curse you before he goes in, and his tools when he has oki are pretty useful (crossup ghost overheads). His damage isn't the greatest with ghosts so don't be afraid, but do recognize that if you get cursed it'll allow Zappa to be covered in more situations. It sucks to hit Zappa to get rid of ghosts, but a flower pot was thrown just before then and hits you for knockdown, and then Zappa wakes up with the sword before you wake up... anyway, look to snipe Zappa with I-No's long range tools, but be wary of doing it when he still has ghosts "equipped" and meter to FRC the ghost throw.

Raoh: Raoh has a hurtbox that makes it easier for I-No to snipe Zappa with HCL, but using it at the wrong time will get you DP'd into a death combo. Play around the DP, but recognize that you have everything you need to blow Zappa up.

What Your Opponent Can Do

Zappa has unusual normals in his "base kit" that are incredibly strong. He also has the ability to summon a dog, sword, set of 3 ghosts, or the hulking thunder god Raoh. They all change Zappa's moveset to an extent and play differently:

Naked Zappa: This moveset is identical to what he gets when he has the dog, except without the dog.
Dog: The dog is commanded with the D button and can act in tandem with Zappa's base moveset, minus his centipedes (the attack that summons). The dog can be used to sandwich an opponent and run lots of mixups and resets, and can take a hit to protect Zappa.
Sword: The sword replaces Zappa's S and H normals and his special moves. The sword moves toward the opponent when commanded to attack and it floats back towards Zappa when left idle or when specific special moves are used.
Ghosts: The ghosts replace Zappa's S and H normals and his special moves. Ghosts have long range normals and can be thrown like projectiles, which will stick to their target and apply a curse on hit.
Raoh: Raoh replaces Zappa's S and H normals and his special moves. Raoh has an incredible DP, good projectile, a super that causes massive chip damage, and very high damage output.