It depends what you mean by "famous." If you mean fame similar to what musicians like Bob Dylan experienced in their early years in the American folk music revival (Greenwich Village, New York City, 1960-1961), a fame of "community awareness/relevance" – then sort of. I mention Bob Dylan for a reason, which you'll see below.
For the most part, Robert Johnson wasn't "famous" during his lifetime, and wouldn't become the blues figure that he is known to be until almost two decades after his death (explained below). The extent of his "fame" when he was alive reached only to a certain range. One, being his circle of friends, (especially musicians he knew, some being famous blues musicians). Two, being those who were involved in the music performance scene (particularly in the Southern States) who came to know him (or of him). And three, others, such as industry professionals and record collectors came to know of him. His "fame" never fell outside of that extent during both his lifetime and for a short while after.
First, you should come to understand that Johnson was a "drifter," or an itinerant musician.
Now well on his way to becoming a polished professional, Johnson established a base in Helena, Arkansas, and worked extensively throughout the South as a walking musician, traveling sometimes alone and sometimes with other guitar players, such as Johnny Shines or Calvin Frazier. He frequently traveled and played under assumed names, a habit that complicated later efforts to construct an accurate biography. It was during this time, between his late teens and midtwenties, that Johnson began to absorb, blend, and refine particular stylistic nuances—drawn from piano as well as guitar—that would eventually help redefine blues for a new generation of musicians who left the South and moved to St. Louis, Detroit, and most prominently, Chicago.
Although Johnson was well known in Arkansas, Mississippi, and Tennessee by the midthirties, he yearned to record, as many of his mentors and influences already had. So, according to most accounts, he traveled to Jackson, Mississippi, to audition for H. C. Speir, a music-store owner whose ear for talent had led to recording sessions for a veritable who's who of important regional blues artists during the twenties and thirties.
Barry Lee Pearson, Billy McCulloch, "Robert Johnson: Lost and Found." University of Illinois Press, 2003.
More interesting is Bob Dylan's account of coming to know about Robert Johnson's music in 1960/1961. From Dylan's Chronicles: Volume One, there is an implication to how Johnson's fame came to elevate in the years after his death.
Before leaving that day, [John Hammond, a record producer for Columbia Records had] given me a couple of records that were not yet available to the public that he thought might interest me. Columbia had bought the vaults of ’30s and ’40s secondary labels — Brunswick, Okeh, Vocalion, ARC — and would be releasing some of the stuff.
This is around 1960/1961.
One of the records that [John Hammond] gave me was The Delmore Brothers with Wayne Rainey, and the other record was called King of the Delta Blues by a singer named Robert Johnson. Wayne Rainey, I used to hear on the radio and he was one of my favorite harmonica players and singers, and I loved The Delmore Brothers, too. But I’d never heard of Robert Johnson, never heard the name, never seen it on any of the compilation blues records. Hammond said I should listen to it, that this guy could "whip anybody." He showed me the artwork, an unusual painting where the painter with the eye stares down from the ceiling into the room and sees this fiercely intense singer and guitar player, looks no more than medium height but with shoulders like an acrobat. What an electrifying cover. I stared at the illustration. Whoever the singer was in the picture, he already had me possessed. Hammond told me that he knew of him from way back, had tried to get him up to New York to perform at the famous Spirituals to Swing Concert but by that time he had discovered that Johnson was gone, had died mysteriously in Mississippi.
I had the thick acetate of the Robert Johnson record in my hands and I asked [Dave] Van Ronk if he ever heard of him. [...] Dave thought Johnson was okay, that the guy was powerful but that it was all derivative. [...] The record that didn’t grab Dave very much had left me numb, like I’d been hit by a tranquilizer bullet.
Johnson recorded in the '30s, and in the 1960s there were still some folks around in the Delta who had known about him. Some even, who knew him. There’d been a fast moving story going around that he had sold his soul to the devil at a four-way crossroads at midnight and that’s how he got to be so good. Well, I don’t know about that. The ones who knew him told a different tale and that was that he had hung around some older blues players in rural parts of Mississippi, played harmonica, was rejected as a bothersome kid, that he went off and learned how to play guitar from a farmhand named Ike Zinnerman, a mysterious character not in any of the history books. Maybe because he didn’t make records.
Bob Dylan, "Chronicles, Volume One." Simon & Schuster, 2004.
You gain some insight and an understanding from reading about Bob Dylan's experience of coming to learn about Robert Johnson. It says a lot about Johnson's (arguably miniscule) fame during his lifetime, and the lack of fame for his work for up to two decades after his death, from before Columbia Records released his compilation album in 1961.
It's also interesting to note, that Dave Van Ronk, who was a prominent figure in the American folk music revival with Bob Dylan, was roommates with Sam B. Charters, a music historian who wrote The Country Blues in 1959. The book, which detailed a whole chapter on Robert Johnson, was written just two years prior to Columbia Records' release of Johnson's compilation album. So, Robert Johnson, John Hammond, Dave Van Ronk, and Bob Dylan were all in some way relevant to one another (which is why I quoted Bob Dylan's mentioning of Dave Van Ronk. Van Ronk knew about Robert Johnson most definitely, especially from his roommate Sam B. Charters).
It was only when Columbia Records released that precise album that Bob Dylan mentions (King of the Delta Blues, 1961), that rose Johnson to supposed "fame," the fame that is accredited to him today. Let me emphasize that it was this album, which was acquired by Columbia Records and released in 1961, that forged Johnson's relevance... two decades after his death.
Mainly, it was friends, musicians/performers, producers (and after his death, record collectors and enthusiasts) who knew, or really, paid attention to Johnson, for the most part. In short, he was not famous at all, not even in the sense of being that much known within "the community," like Joan Baez was, for instance, in the American folk music revival prior to Bob Dylan's fame.
Robert Johnson first entered a recording studio in November 1936, and he died in August 1938. His recording of 'Terraplane Blues' for Vocalion Records - released in his lifetime - is usually claimed to have been a minor regional hit in books like Elijah Wald's Escaping The Delta, while other singles released before his death were less successful. However, Billboard didn't start polling record retailers about sales until 1940, so it's hard to verify.
Of course, recordings in the 1930s weren't as big a contributor to a musician's success as they are today, though they certainly didn't hurt. Johnson was clearly well-known in the delta blues community - he was an itinerant working musician, and so fans of the music in the areas where he lived would have known his name. Certainly plenty of the delta bluesmen who survived until the 1960s, like Son House, knew who Johnson was and told stories about him. /u/shy mentions John Hammond's From Spirituals To Swing concert, which I discussed more here; this was a big boost to the careers of some of the musicians who performed, and might have also been so for Johnson had he lived for a few more months.
However, the important context for Johnson that is less well-understood today, but which is clearer in books like Wald's Escaping The Delta is this: Johnson's delta blues style, in 1937, was already old-fashioned. The heyday of the delta blues style we associate with him was in the late 1920s. Johnson recording music in that style in 1937 is broadly equivalent to someone releasing music in 2016 that sounds like the music of 2005-2009; say, the big hits of the Black Eyed Peas or Fall Out Boy.
The record industry had discovered that black Americans bought records in the mid-1920s, and so the late 1920s saw a rush to record the popular music style of the time in the South, where people were buying records - this happened to be the delta blues. However, the onset of the Great Depression sent a great many record companies bankrupt. Other record companies de-emphasised putting out music recorded mostly for regional ethnic minorities (as the record industry generally regarded black people in the South). Because of this, and because of the general economic downturn, most of the delta blues musicians recorded in the late 1920s - including some of the most popular - had given up on music as a career by the early 1930s, going back to occupations like sharecropping. Such musicians, if they survived until the 1960s, were often re-discovered by intrepid white fans and encouraged to play folk festivals and the like.
Vocalion, the record label Johnson recorded for, was relatively unusual as a record company in that it survived the depression while still putting some focus on black music. However, it's instructive that there aren't a great many delta blues artists on Vocalion's catalogue circa 1936-1937 - instead the record label at the time seems to be putting more emphasis on swing and jazz acts like Fletcher Henderson and Louis Armstrong, presumably because this is what the black community was most likely to buy at the time. All in all, it's probably unusual that Robert Johnson got a chance at all to record delta blues in 1937, and it was probably not surprising it wasn't that great a hit.
Well if there's no blues historians here, I'll give it a crack if the mods are okay with it. My only qualification is living in the Delta.
Robert Johnson, like many poor black men of his time, doesn't have much documentation following his life. His birthdate is even up for debate, but is assumed to be in May of 1911. It is assumed based on what documents we do have of his life, like his marriage certificate. It is also known he was considered a lackluster guitarist until he was a bit older when he left his home town and came back a few months later seemingly having become a master overnight, lending to the myth of him selling his soul to the devil.
Shortly after this, he began moving from city to city making a living playing wherever he could, often under aliases as he did not want to be known. His popularity rose in the 60s when his recordings were discovered and his genius could be appreciated. So while he created quite a buzz in each place he went, he was far from a superstar.
Apologies and thanks to the mod team for running a tight ship. The Blues Foundation offers a brief description of him and his life: http://www.robertjohnsonbluesfoundation.org/biography/
Bio.com is far from comprehensive but provides backing for the claims of his relative obscurity during life and a couple of more documentaries about his life: http://www.biography.com/people/robert-johnson-9356324#death-and-legacy
The Rock and Roll Hall of Fame has a decent piece on him. However, as you might be able to tell by now, there's much conflicting information on his life. This is no less true in this source: https://rockhall.com/inductees/robert-johnson/bio/
I did, however find a YouTube link to the documentary I previously mentioned. I feel like sourcing YouTube is a bit lame, but out of all the sources, I would trust this one most personally. Robert McCormick and John Hammond have dedicated their lives in the search of documenting the legacy of old legends of American music. https://youtu.be/ONZbSir45rQ