[deleted]
There's some ambiguity about the question-- when you refer to the "early 1900s" and then refer to "merchants travelling across the land" -- it sounds as though you're talking about two entirely different time periods.
In the early 20th century-- indeed since 1874-- international postal services have been coordinated under the Universal Postal Union [UPU], established by the Treaty of Berne in 1874^([1]). The UPU still exists today and is a "specialized agency" of the United Nations^([2])
This treaty regulates the way in which mail moves from one postal service to the next; that's not "merchants traveling across the land", but rather entirely a bureaucratic process. Mail enters a national postal system and if addressed to a foreign country, is transferred to that other nation's postal system for delivery. So you mail a letter in Indianapolis for delivery to your cousin in Amsterdam. The mail travels in the US system to an international transfer, arrives in Europe, and enters the Dutch system for delivery.
In much earlier times, mail was handled as you suggest in ad hoc arrangements-- bundles of letters from a ship travelling West might be exchanged with letters travelling East, but that's not the "early 1900s".
National postal systems were established centuries ago-- when the United States Constitution was written, the Federal Government is given authority to "To establish Post Offices and post Roads”^([3]) the following year 1789, Congress passes a law giving the President the power to appoint a Postmaster General^([4]); President Washington appoints Samuel Osgood as the first Postmaster General.
If what you're looking for is a general introduction to the history of post offices and correspondence, you've got a tremendous span of history. We have business correspondence written in cuneiform on clay tablets from four thousand years ago. Here's an ancient Assyrian letter (actually a clay tablet, the writing is a kind of scratch baked into the clay) demanding payment on a debt, and making reference to how previous demands would have been delivered-- by caravan.
A message from Silla-Labbum and Elani: Tell Puzur-Assur, Amua, and Assur-samsi:
Thirty years ago you left the city of Assur. You have never made a deposit since, and we have not recovered one shekel of silver from you, but we have never made you feel bad about this. Our tablets have been to to you with caravan after caravan, but no report from you has ever come here.^([5])
So in summary: thousands of years ago -- and even hundreds of years ago-- letters might move "across the land" carried by merchant caravan, finding the intended addressee by informal means. But that's not the "early 1900s", at which time there was a well established national and international postal system. Because it was so important for business, physical paper mail moved very fast in the early 20th century-- there were often more than one mail delivery per day, as many as a dozen in Victorian London; a correspondent might request a reply "by return of post", meaning that you'd get the mail in the morning, write your reply and put it out for return the same day^([6])
Finally there's the "how do you know it got to the right person" problem. Modern postal systems have registered and certified mail, mechanisms by which a sender can prove that a letter was either delivered or delivered and signed for by a named person. Systems to prove such things are quite old-- in the time of Mary Tudor (1556), we find
the poste between this and the Northe should eche of them keepe a booke and entrye of every letter that he shall receive, the tyme of the deliverie thereof unto his hands with the parties names that shall bring it unto him, whose handes he shall also take to his booke, witnessing the same note to be trewe^([7])
In ancient Assyrian example, business correspondents transmitting tablets by cuneiform known to each other might ask the caravan driver -- "hey, did you give my packet to Puzur-Assur?"
[1] Treaty of Berne, General Postal Union; October 9, 1874
[3] United States Constitution, Article 1, Section 8
[4] Act of September 22, 1789 (1 Stat. 70)
[5] Letters From Mesopotamia: Official, Business, and Private Letters on Clay Tablets from Two Millennia, Chicago: University of Chicago Press, 1967
[6]Smithsonian National Postal Museum
[7] The history of the post office from its establishment down to 1836