FBI SA, George Franklin, during nightfire training circa 1935 (photo courtesy of his son)From our own research, in all liklihood the FBI's first purchases of the Thompson sub-machine gun ocurred in June, 1933 soon after the Kansas City Massacre. One can see from other documents in the "navigation" area that Hoover's Committee had decided on the use of the Thompson and other weapons earlier in the month of June, 1933.
The first two Thompsons obtained by the Bureau, revealed in these documents, were no doubt sent to Kansas City FBI as a result of the "appeal" for weapons made by SAC Ralph Colvin which you'll see in the "navigation" area. The Bureau also purchased spare parts for those two Thompsons.
The supporting record for this first purchase of 2 Thompsons is here.
The first large purchase of Thompsons is believed to have been at the end of June, 1933 as reflected by the document revealing a purchase of twenty Thompsons for distribution to the Field Offices.
You can obtain the Thompson documents mentioned above here.
I also suggest you read the Thompson article by Bo Ramsur which is located in the "Articles" Section of the site.