Who were the first GMs, and how did they become GMs? In order to qualify for the title of Grandmaster, a player must achieve three Grandmaster norms. This is not the place to go into detail about what those are; the pertinent point is that norms can only be gained in tournaments that fulfill certain criteria, including the requirement that the entry must include at least three Grandmasters.

Therefore, it seems to me that it's impossible to become a Grandmaster if there weren't already at least three Grandmasters in existence. How did the first Grandmasters earn the title, and who were they?
In order to qualify for the title of Grandmaster, a player must
  achieve three Grandmaster norms.


That is not accurate. The regulations for the award of FIDE titles are given here.

There are two ways to become a grandmaster. Achieving norms and reaching a certain rating is one way but there are also Direct Titles. The FIDE handbook defines these like this:


  Direct title (automatic title) is a title gained by achieving a certain place or result in a tournament. For example, winning, or
  achieving a result ≥50 percent in a tournament. On application by the
  player’s federation and confirmation by the Qualification Commission,
  such titles are awarded automatically by FIDE.


The tournaments and achievements which qualify the player for the various titles are given in the Table for Direct Titles effective from 1 July 2017.

For the grandmaster title the following results give the player the right to the GM title:

Last 16 of the World Cup
Winner of the Women's World Championship
Outright winner of any of World S65, S50, U20 championships
Outright winner of Continental Individual Championship

But in any case these regulations are made by FIDE and they change with time. 


  Who were the first GMs, and how did they become GMs?


The original FIDE grandmasters were awarded the title via a resolution of the FIDE general assembly in 1950. These players were:


The top players of the day: world champion Mikhail Botvinnik, and those who had qualified for (or been seeded into) the inaugural Candidates Tournament in 1950: Isaac Boleslavsky, Igor Bondarevsky, David Bronstein, Max Euwe, Reuben Fine, Salo Flohr, Paul Keres, Alexander Kotov, Andor Lilienthal, Miguel Najdorf, Samuel Reshevsky, Vasily Smyslov, Gideon Ståhlberg, and László Szabó.
Players still living who, though past their best in 1950, were recognised as having been world class when at their peak: Ossip Bernstein, Oldřich Duras, Ernst Grünfeld, Boris Kostić, Grigory Levenfish, Géza Maróczy, Jacques Mieses, Viacheslav Ragozin, Akiba Rubinstein, Friedrich Sämisch, Savielly Tartakower, and Milan Vidmar.


In 1951 Efim Bogoljubov was added to the list via another vote.

In 1953 something similar to the modern rules was introduced with the world champion and players who qualified for the candidates having the right to the title (if they weren't already) being joined by a variation of the current norm based qualification.

These rules received major revisions in 1957, 1965 and 1970 and may have been tweaked on further occasions.