lostuser the Play Integrity API is used by banking apps to verify that the app is running on a genuine, unmodified device and that it was installed from the Google Play Store
This is technically true (that's what they use it for), but deceiving. Play Integrity is misnamed, it's actually checking certification, not integrity. GrapheneOS is very pro-user, so they aren't and wouldn't be certified by Google.
The correct feature, supported in GrapheneOS, fulfilling the same purpose is attestation.
lostuser This helps protect against fraud and unauthorized access to sensitive financial information.
No. It's not an integrity check.
lostuser Are there any downsides, especially regarding security and privacy, to access my bank account and carry out financial transactions via the web browser (Vanadium) instead of the official app available on the Play Store?
Yes. Official apps can be hardcoded to recognize the certificate of their servers, for more secure connection authenticity. Google Play Store itself does this. It's also available from the built-in App Store, which also does this. And the built-in App Store is protected by verified boot, so there's a strong chain of security from GrapheneOS to the App Store to Google Play Store to your bank app to your bank's servers.
lostuser Does Vanadium, or any other browser for that matter, offer features similar to Google’s Play Integrity API
No. The web is meant to be an open platform. I'm not aware of any attestation API available for websites, I think it was actually proposed in the past and met with backlash (rightfully, in my opinion). But this means that the web is less secure. You have weaker authenticity when connecting to websites, and websites can't use attestation to verify your hardware and software. (The web browser can be more secure though, because the web browser has its own sandbox in addition to Android's sandbox, which provides an additional layer of defense, and websites have a much reduced visibility of your system and features exposed to them, which reduces attack surface.)