The most likely explanation based on the storyline is that Andy, having escaped from Shawshank just the night before and not knowing the extent or progress of any efforts to find him and return him to prison, prioritized completing all local business as quickly as possible, especially the most public tasks (he could save for later things like planting the note and money for Red in the agreed upon location), in order to maximize his chances of making it out of New England and onward to Texas and then Mexico, where he could live essentially free of the risk and fear of being recaptured by authorities. At that point in the story, Andy was already in the middle of decisively playing his extremely risky hand, and he added no additional risk by leaving the package in the hands of the bankers, who would certainly remember him and his visit regardless of that choice, whereas even taking the time to go to a post office would have added precious minutes to his public appearances in the area and exposed him to more people.
The filmmakers' goal of efficient and thorough storytelling provides several more probable explanations, the most straightforward being that the choice allows viewers to see the mailing explicitly by adding only a few seconds to the bank scene. To accomplish that same storytelling goal, another means, such as a trip to the post office, would have likely required a whole extra scene that might have unnecessarily broken up the flow of the movie in a crucial phase of its story and certainly would have added run-time to the film, as well as an additional shooting location, perhaps more actors, time, cost, etc. to the film's production. Andy's decision to allow the bank to handle the mailing, perhaps unplanned and prompted only by the female bank employee's asking him politely whether they could "do anything else" for him, gave the filmmakers an opportunity not only to show the mailing itself but also to highlight, through the casual and gentle attitude he displays, Andy's confidence and aura of tranquility, remarkable (but not too surprising at this point of the film, coming from Andy) given the perilous nature and the extremely high stakes of his situation, not just for him personally but also for countless others and perhaps even for society as a whole. The brief exchange also emphasizes the dramatic shift of power almost two decades in the making that Andy had finally, dramatically brought about and was, at that moment, pointedly experiencing for himself[spoiler], having broken away from the cruel and domineering rule of Shawshank and Warden Norton and finding himself no longer society's prisoner and Norton's slave but instead the deeply respected and important customer of Norton's bank and, more importantly, the recipient of Norton's tainted fortune and the server of proper justice regarding the ironic criminal activities of his former prison.