I'm planning a banking guild that aims to provide a wide range of services within finance, consultation, information, contracts, insurance, and estate. But please understand that everything in Elyria will be player driven, so as GranDad1982 stated having bank accounts that can be accessed from everywhere is simply not possible.
Transfer of money by receipt isn't impossible however, but not necessarily as an account service. An example could be: 1. You pay in an amount plus a fee to Bank A and receive a receipt. 2. You can then use that receipt at Bank B for a fee where they pay you from their own reserves. 3. Bank B would then request a compensation from Bank A who would send the amount you paid them with a courier to Bank B.
This can be seen as a travel insurance for the client, hence the fees on both ends, but don't expect a modern accounting system.
Personally my bank will specialize in investments rather than accounting, to ensure your money keeps circulating and growing. The closest thing to a traditional bank account in that regard would be to invest in debtor contracts.
For regular materials and such I will recommend renting or acquiring a storage building though. If my bank ever offers a form of accounting service it'll be in the form of mini-vaults and safes where the client keeps the sole key to avoid trust issues. Those will primarily be ment for long-term storage of trinkets, documents, and other valuables though, not for short-term storage of materials, consumables, or currency. Thus expect deposits and withdrawals to be pretty expensive compared to the annual rental fee itself.