Is the process of purchasing shares through an Islamic bank and speculating with them—in the shares of companies that have been deemed permissible for speculation—after the bank delivers a letter to the brokerage firm to buy these shares, a permissible act?

There is no objection to dealing with Sharia-compliant Islamic banks, nor to purchasing something from them and then being referred to an entity that deals with Sharia-compliant transactions to receive it. One can also consult the Sharia committees of Islamic banks for detailed information.