National Postal Museum, mail chutes like this were introduced in the late 19th century. Here's a bit more:

Cutler's company was the sole manufacturer of mail chutes and receiving boxes until 1904. During those 20 years, the company produced more than 1600 receiving boxes, and continued to produce them for several more years. This receiving box was constructed in 1920.

This page suggests you can still buy mail chutes, but not Cutlers. So that means this mail chute is at least 84 years old? It's funny, because the exterior of the building makes it look fairly modern. I guess that's what you call a facade. Deceptive appearances in the heart of America's capital!? Shocking, isn't it?