Can I prove my app uses randomness right way? We are developing an iOS app which employs Shamir Secret Sharing algorithm. We made all possible steps users can trust us. We can't remember their secrets nor we can save encoded fragments. But they still may not trust us with randomness. Is there any cryptographic method how user can verify we are using random number generator honestly and have no malicious interests?
Nothing you show the user within the app could prove that the numbers were truly generated randomly and that the numbers weren't leaked maliciously or accidentally.

In general, you could make your application (or the module that does the cryptography, assuming it's built into a separate binary like a library or an executable used by the main app) open source and use a reproducible/deterministic build process so that users can compile the source themselves and verify that it matches what they've installed. I'm not sure how feasible or convenient this may be on the iOS ecosystem though. (Users might need a rooted phone to be able to view the app's binary installed from the app store, etc.)