I am reading your questions as asking "Is there any basis for a story of Sargon of Arkad?" but I am assuming you want to know not just whether such a basis exists, but something about its reliability. In other words your comment about "whether it's true or not" refers to the question of whether there is a relationship between this story and that of Moses.
The short answer is, yes, there is a story, usually referred to as the "Sargon Birth Legend". It is known from 4 fragmentary sources (A, B, C and D - the letters in the left hand column of Sklivvz's answer refer to these sources -- see Westenholz. Page 41.).
They are written in neo-Assyrian and neo-Babylonian. Obviously this does not precisely date the tablets. The most thorough study of the dating that I am aware of is in Lewis. Unfortunately I do not have a copy of this, but Konstan cites him as giving a date range of 704 - 539BC. These are not the oldest materials we have concerning Sargon's life - there are a few with dates as early as 1800BC (see Konstan, page 27).
As Longman points out (page 54) there must be an element of fiction in the account because it is presented as autobiographical but Sargon says at one stage "I ruled 55 years as king".
Longman describes the state of research on the story (as of 1991). It seems clear that no scholars cited by him believe the story to be original to Sargon or his time period, but that it is a much later invention and I know of no-one that thinks otherwise.
Lewis's view was that the story was composed at the time of Sargon II of Assyria (an unrelated monarch, who reigned in the 8th century BC, but may have been named in honour of Sargon of Akkad) to reflect glory on him indirectly.
But it wasn't entirely clear to me whether this was your question, or whether you wanted to know if the story was popular (as is the story of Moses) rather than whether it has any basis in fact.
Clearly it was not popular for much of history because knowledge of Sargon was lost, but it does seem that Sargon was a very significant figure in Mesopotamian history to the Mesopotamians. Babylonians and Assyrians certainly thought of themselves as in some sense his successors (as the first Semitic king). So, in that sense, yes it was a popular story at some time in the past.
The question of whether this story and the Moses story are related and, if so, how is much more difficult and not one you asked. I am just flagging up that I am not trying to answer it.
Konstan, David, and Kurt A. Raaflaub, eds. Epic and history. John Wiley & Sons, 2009.
Lewis, Brian. The Sargon legend: a study of the Akkadian text and the tale of the hero who was exposed at birth. American Schools of Oriental Research, 1980.
Longman, Tremper. Fictional Akkadian autobiography: a generic and comparative study. Eisenbrauns, 1991.
Westenholz, Joan Goodnick. Legends of the Kings of Akkade: The Texts (Mesopotamian Civilizations). Eisenbrauns, 1997.