Although there is documented evidence showing that the FBI purchased Thompson sub machine guns after the Kansas City Massacre in 1933, it is also clear that on some occasions Field Offices were still lacking in "firepower" and had to make emergency borrowing requests for weapons.
In this instance, Dillinger and his gang were believed to be in the Indiana area in April, 1934 and SAC E. J. Connelly, with Clyde Tolson's approval and assistance, borrowed multiple weapons from the Federal Laboratories in Pittsburgh, Pa. These guns etc. were transported by 2 Pittsburgh Agents by train and delivered to Connelly.
Two April, 1934 documents show the "borrowing process" and the details of the transport.
You can also see in the first document that Clyde Tolson is "ordering" 30 additional Thompsons and other rifles to assist FBI offices around the country.
You can see one document here relative to a request by borrow weapons with regard to the anticipated problems the Agents were going to have with the Dillinger gang.
You can see the delivery of those weapons to E. J. Connelly here.