I know for p2p they have a beast of burden thing, is there something similar for f2p? Like an amulet or something?
The closest thing to automated banking for f2p players is magic notepaper which can note items in your inventory and legendary pets from Solomon's store which can bank items every 10 minutes. These do not automatically bank items.
Beast of burdens also do not automate banking. The main p2p thing that automatically banks items are sign of the porters.