edit: One note about something that was bugging me for a while...items created on my computer sometimes wouldn't show up in the vault for immediate use. Painful when you sign up for a service using your computer and then try to immediately sign into it on your phone.
In the iOS app settings there is 'Swipe down to refresh' (or similar) - turn that ON. Not sure why it was off by default, but it totally fixes the issue. Just swipe down to refresh the vault and your new item appears.
Having a push feature only works if you can engineer your app or add-in to open up the necessary ports or tunnels in the OS itself. Polling on the client end will always be easier to implement.
Also: just checked BitWarden v2.8.0 (449) on iOS 14.4, no setting for “swipe to refresh” anywhere in its settings.
My trouble was specifically related to the 'Pull down to refresh' behavior being disabled by default though. If that feature is disabled the new items will appear sometime, with no way of knowing when that will be. I honestly don't even know why that feature has an ON/OFF switch, it should just be permanently enabled.
Settings -> Manage -> Sync -> Enable sync on refresh