If the app's open-source, you can build it yourself, with restrictions on how long the app will function on your device before you have to reinstall it (one year IIRC for paid developer accounts, but only 7 days if you're not enrolled in the paid program).

If the app isn't open-source, you don't have any great options. Hypothetically, compiled apps from other developers can be re-signed just like apps you build yourself (see AltStore [1], which uses this technique), but those apps are still time-bombed and have to be periodically refreshed. The barrier to entry means you don't see a ton of apps around that do this; if something can't be compliant with App Store rules, it generally doesn't get made on iOS.

(There's also jailbreaking, and there used to be a decently-large community of developers building applications and tweaks for jailbroken iOS devices. That's gotten smaller both as Apple's made jailbreaking more difficult to maintain, and as new features in iOS have made much of the functionality people used to jailbreak for redundant.)

[1] https://altstore.io/