Hive works in the browser exactly the same as in the vm. The only difference is how your data is stored. Since the browser has no file system, Hive uses IndexedDB as storage backend. Every box is stored in its own IndexedDB database. You don't have to care about that, everything works as usual.
You should be aware that it is easier for the users to view or change your boxes in the browser than on mobile phones (for example using Chrome DevTools). You can use an encrypted box and nobody will be able to read it.
TypeAdapters work as usual.