Retro games and Apple devices are two of my main tech interests. A lot of people, including myself for the longest time, think that there is no way to emulate games if you use Apple devices unless you go through the frustrating process of loading AltStore/jailbreaking. However there are plenty of emulators that run via a web app that can be installed via safari. The essential iOS app thread inspired me to make this thread since these didn't really fit the criteria of an "iOS app". I'm going to list the sites and platforms that I use but please post your preferences in the comments.
Afterplay is my personal favorite. It can play SNES, GB/GBC/GBA on the free tier. NES, Genesis, and DS support is part of the premium tier ($6 a month). Whether you are on the free or paid tier, this is the only web emulator that I know of that has true save and ROM syncing between devices. Afterplay uses lib-retro cores as its backend and exposes a lot of different options that other web emulators don't. A major con of this project is that it doesn't have offline support at the moment but it has been announced as coming.
DS Player is a DS emulator that uses desmume-wasm as its emulator core. It allows you to connect to dropbox to sync saves. It features offline support but is a lot more barebones than afterplay's ds core. Its totally free however
Emulator.JS is another project like after play in that it is a web frontend for a lot of js/wasm ports of lib-retro cores. It runs a multitude of systems including N64 and PS1 and it runs them at actually playable speed. However I can only seem to get it to work with touch controls on iOS. I don't know if that is a problem with my controller (I use a razor Kishi 2) or with the emulator.js. It recognizes the controller just won't rebind the controls no matter what I do.
[Eclipse](eclipse emu.me) is a web emulator with focus on NES, SNES, and GB/GBC/GBA. It allows you to connect to google drive and manage your library via the cloud. It also has offline support and is completely free.
Mods, I'm not involved with any of these projects so I'm I'm sorry if this breaks rules on advertising.
If you don’t have the hardware to dump your games yourself, then googling the game name and then archive.org is a good bet to get results.