Depends on what you mean by "join Axis". Historically, Italy joined (technically they helped create) the Rome-Berlin-Tokyo Axis in October of 1936, although this was mostly a symbolic "alliance" at the time. So to was the Anti-Comintern Pact of the following month (which Italy signed in 1937). It didn't become a true Military Alliance until May of 1939 with the "Pact of Steel" and the full Axis in September, 1940 with the Tripartite Pact. However, Italy joined militarily in June 1940 (i.e., before the Tripartite Pact but after the Pact of Steel) when it was apparent that France would fall.
So in game, the most accurate restriction would be that Italy can join the alliance whenever, but can't be called into the war until German troops have crossed the French border, either through the Low Countries (historical), through Switzerland (Operation Tannenbaum), or by breaking the Maginot Line.
As a note, I am pretty sure the game is coded that Germany can't (EDIT: Won't} call Italy before then, but its allies (including Slovakia and Hungary) can, which leads to Italy joining the war early. As a player I never call Italy (if I even bother to ally them) before the fall of France.