Apple just added a few lines to their iOS App Store review guidelines, including this:
“Additionally, retro game console emulator apps can offer to download games,"
This seems to imply that retro game console emulators (the PSP is almost 20 years old, that's has to count as "retro") will be allowed in the iOS App Store - however, it's also mentioned later that "links must be provided to all downloadable software", which makes it unclear if it emulators will be allowed to let the user pick from their own files, and not just app-internal downloads.
Since we don't own the rights to PSP games, we can't offer them as in-app downloads, users must still obtain the games on their own (by dumping UMDs). So, for PPSSPP to be useful beyond running a small set of free homebrew games, it all depends on how Apple interprets their own rules.
If it turns out that the rules now allow emulators with ISO/ROM pickers, PPSSPP will come to the App Store later this year. If not, well, it won't!
(JIT permissions have also been debated, and don't look likely to be approved. However, PPSSPP doesn't need JIT to perform well on modern Apple CPUs.)
The current status of the iOS port
UPDATE (2024-04-29)
The successful release of the Delta multisystem emulator seems to confirm that proper emulators can now pass App Store review. In the near future, an attempt will be made to submit PPSSPP to the App Store.