The claim is about the early days of computing, and there are two parts: that the petroleum industry was the initial lead consumer, and that the federal government eventually replaced it. We can look at the first two large machines sold commercially in the US: the Univac I and the IBM 701.

According to the list at http://en.wikipedia.org/wiki/UNIVAC_I there were 18 Univac 1 installations. None went to petroleum firms. The first 6 all went to the Federal government. 3 went to insurance companies and several to utilities.

The IBM 701 had 18 installations. http://www-03.ibm.com/ibm/history/exhibits/701/701_customers.html None were for petroleum companies. Only three seems to have been directly for the federal government. Eight were for aerospace firms (and so supported by Federal contracts indirectly). A couple went to universities.

The IBM 702 customer list is less easily accessible, but I suspect would be dominated by utilities, large manufacturers, and insurance firms.

So neither claim holds up for large machines of the first generation. Of course by number the IBM 650 sold far more than any of the other models, but it was by no stretch of the imagination a large number cruncher.

According to the Feb 1958 issue of Office Automation, of 472 IBM 650 computers for which data was available (from a population of around 1,100) the biggest user sectors are refinery (9.8%), Aircraft (8.9%), Insurance (8.8%) and electrical and electronic manufacturing  firms. (8.4%) About 12% were in the federal govt and military.

So the claim holds up much better for small computers in the mid-1950s – refineries are the biggest private sector users, though still not literally true as the Federal government is larger still.

Tom