What familiars in Runescape will bank items for you?

Solution 1:

There is only one familiar which can deposit items from your inventory to the bank. The Pack yak. It requires 96 summoning and, with a Winter Storage scroll, you can bank an item from anywhere.

However, there are three pets from the Solomon's General Store which have an ability to bank items. Note that these pets cost 1300 runecoins for free to play players, or 1170 runecoins for members.

Although this will not help you, the abyssal titan has a special move which banks all essence from its Beast of Burden and your inventory to the bank. However, this is only limited to essence, so it won't help with your steel bars. It also needs 93 summoning to summon and the scroll uses 6 points.

Solution 2:

Only the Pack Yak has the 'Winter Storage' ability to send it's inventory to your bank. The Abyssal Titan can send all pure and rune essences in your inventory and it's own to the bank with it's 'Essence Shipment' ability. No other familiar abilities send items directly to the bank.

Source: https://runescape.wiki/w/Summoning_familiars