Ironically most of the titanium used to build it was supplied by the USSR, as one of the leaders in titanium production. The other major supplier is South Africa, mined out of dunes, where it is a black impurity in the sea sand.
This is part of Russian propaganda. It is common in Russia, as it was in the USSR, to create myths about Russian greatness. When the country lacks a promising future and has little to be proud of in the present, they glorify the past, often fabricating stories.
SR-71 production began in 1962 and ended in 1968. During this period, about three quarters of the US titanium sponge demand was met by domestic production, and only one quarter was imported. Imports from the USSR began in 1965, three years after production of SR-71 started, accounting for 2% of titanium sponge imports (0.5% of demand) in 1965-66, rising to 19% (5% of demand) in 1967.
Soviet titanium sponge was traded at lower prices than domestically produced titanium and imports from the UK and Japan, the main trading partners for titanium. Price undercuts is the only reason for growth of their imports.
There is no evidence that military and government contractors sourced titanium for their projects directly from USSR, there was no need for that, and chronologically events do not align.