Earlier quoted context omitted.
They could probably sue Apple, too. When someone buys something on the App Store, the store makes a copy and distributes a copy to the purchaser. If that copy is infringing, then the entity that makes and distributes the copy (i.e., Apple) should be liable. That they received their initial copy from someone else (i.e., Automatic) who told them that they had the rights to make and authorize making and distribution of…
IANAL but I don't think Apple is liable in this case. It sounds just like websites not being responsible for user created content. They do have to remove illegal content, and must have means to report and act on it, but you can't e.g. sue hackernews because someone posts a torrent of a Marvel movie.
Apple might not know it is infringing, and indeed might even think it has permission because the developer told it the developer had made sure all the licensing was in order, but the defendant's mental state in a case of direct copyright infringement is not relevant. Copyright infringement does not have a mens rea, at least in the US. "[i]ntention to infringe is not essential under the Act.", Buck v. Jewell-Lasalle Realty Co., 283 U.S. 191, 198 (1931).
The website shield against liability for user created content that is most often used is from section 230 of the CDA. That specifically carves out an exception to the shield for anything relating to intellectual property, so 230 is out.
The DMCA looks like it might work. The people claiming infringement could send a DMCA take down request to Apple, and Apple could take down the content and give them the contact information necessary to sue Automatic, and that should get Apple out of the loop.
But DMCA has limitations, too. If a service provider is infringing themselves, or acts in concert with the people providing the infringing material, then the DMCA safe harbors may not apply. Acting in concert is more than just providing hosting, and I have no idea if Apple does enough more to reach acting in concert. There's some interesting Q&As about DMCA safe harbors at https://assets.fenwick.com/legacy/FenwickDocuments/DMCA-QA.p...