Hi there, there is no way currently to backup/restore Vanadium app data. For more information about why that is, and whether it will change in the future, please see this bug report in the Vanadium issue tracker on GitHub: https://github.com/GrapheneOS/Vanadium/issues/102.
There are very few options in the current implementation of Vanadium. You can export your browser tabs by selecting them all, and sharing them with notepad or your clipboard. You can also export any passwords you have saved. It's not possible to do the same with bookmarks or browser history. There is also no way to import the browser tabs and passwords you saved onto your new device other than opening them one by one.