Zotero's own solution to using the program in your tablet browser is the "Bookmarklet," a special bookmark you can add to your browser that will allow you to save a page you are reading to your Zotero library.
You must have a Zotero account, and your tablet must be online in order to use Bookmarklet.
Now, when you're on a page with a citation, you can use the Save to Zotero bookmark to save the citation directly to your Zotero library online.
While there is no official Zotero app at present, Zotero's website lists a number of third party apps that work with Zotero.
Zotero Reader (allows you to view your Zotero library, including attachments, in Android, iPad and desktop)
BibUp (allows you to add books to your library by scanning their ISBN barcodes)
PaperShip (allows you to view and add to your library, read and annotate attachments, and share references)
ZotPad (allows you to view and edit your library, as well as read and annotate attachments).
Scanner for Zotero (allows you to add books to your library by scanning their ISBN barcodes)
Zandy (view and edit your Zotero library)
Zotfile (a Zotero plugin that syncs your library with your mobile PDF reader and extracts mobile annotations into Zotero notes)