I'm searching around, and I can't find any sources that make the claim that the Soviet Union provided food aid to Americans either in the Dust Bowl, or the Great Depression more generally. I would actually be curious where and who is making this claim.
I can provide a little context for where Soviet - US relations were in this period. The United States had withheld official diplomatic recognition of the USSR from its founding until 1933. While much of this was due to the bad blood around US and foreign intervention in the Russian Civil War and the 1919 Red Scare in the US, a major stumbling block was the issue of unpaid tsarist-era debts - the Soviets refused to honor them, and the US insisted on repayment to bondholders.
The new Roosevelt administration was open to re-establishing diplomatic relations, however, and did so after a series of negotiations with Soviet Foreign Minister Maksim Litvinov in November 1933 (the issue of debt repayments was pushed off to be settled at a later time). William Bullitt, who had led a negotiating team to talk to Lenin on Wilson's behalf in 1919, assumed the post of US ambassador in December.
So right off the bat, I can't imagine that there would be any sort of aid traveling from the USSR to the US before December 1933, and thus not for the worst parts of the Great Depression.
But in addition, Soviet foodstuffs were not something that would be easily parted with. In no small part this is because the USSR had its own issues with food supplies, especially in 1930-1934, when famine conditions killed something like 5 to 7 million people across the USSR (especially in Ukraine, where it is known as the Holodomor). Much of the grain that the state did collect was for use as exports, as the Soviet government relied on selling grain on the international market in order to earn the foreign currency it needed to pay for imports of industrial materials and to pay for contracts with Western firms for the development of industrial facilities. Indeed, starting in 1929 the Soviet government had already signed agreements with such companies as Ford Motor Company, Caterpillar and DuPont for the construction of industrial production facilities in the USSR.
While Russia had been the largest single exporter of wheat until 1914, during World War I, the Revolution and Civil War, wheat production had collapsed - the American Relief Administration was in fact allowed by Lenin's government to provide aid in the country from 1921 to 1922. Soviet wheat exports began to recover on the international market after 1926, and rose from 1.3% of total exports to 7% by 1931, but collectivization, famine and political crisis largely halted this growth. In the meantime, the major international wheat producers of the Interwar period - the United States, Canada, Argentina and Australia - accounted for some 90% of total exports. With the economic downturn, there was, if anything, a world glut in wheat and prices fell (this was particularly hard on the Soviets, as they continuously needed to sell more wheat to earn a steady amount of foreign currency).
Now, US production of wheat in particular was hard hit during the Dust Bowl and associated drought conditions on the Great Plains - the Department of Agriculture provides the following data for winter wheat:
| Year | Production (1000s of bushels) | Price per bushel ($) |
|---|---|---|
| 1915 | 640,565 | na |
| 1916 | 456,118 | na |
| 1917 | 389,956 | na |
| 1918 | 556,506 | na |
| 1919 | 748,460 | 2.10 |
| 1920 | 613,227 | 1.48 |
| 1921 | 602,793 | 0.945 |
| 1922 | 571,459 | 1.04 |
| 1923 | 555,299 | 0.945 |
| 1924 | 573,563 | 1.32 |
| 1925 | 400,619 | 1.48 |
| 1926 | 631,607 | 1.21 |
| 1927 | 548,188 | 1.16 |
| 1928 | 579,066 | 1.03 |
| 1929 | 587,057 | 1.04 |
| 1930 | 633,809 | 0.694 |
| 1931 | 825,315 | 0.382 |
| 1932 | 491,511 | 0.391 |
| 1933 | 378,283 | 0.777 |
| 1934 | 438,683 | 0.844 |
| 1935 | 469,412 | 0.827 |
| 1936 | 523,603 | 1.02 |
| 1937 | 688,574 | 0.977 |
| 1938 | 685,178 | 0.574 |
| 1939 | 565,672 | 0.694 |
| 1940 | 592,809 | 0.69 |
| 1941 | 673,727 | 0.957 |
| 1942 | 702,159 | 1.11 |
| 1943 | 537,476 | 1.39 |
| 1944 | 751,901 | 1.43 |
| 1945 | 816,989 | 1.50 |
You'll have to excuse this data dump - I mostly just want to show where wheat production hit its low in the mid-1930s, and show how prices collapsed with the onset of the world Depression. The Dust Bowl is connected with these events, but for what its worth that is more specific to droughts in 1934 and 1936, and major dust storms in 1935. It's worth noting that the US went from making up a quarter of world wheat exports in 1930 to a tenth of exports during the 1930s, and during the worst of the Dust Bowl years was a net importer, importing some 3.9 million bushels of wheat during 1934-1935 - but was able to import wheat nonetheless. In fact, attempts by a 22 countries to stabilize wheat exports and prices through price stabilization measures and export quotas under a Wheat Advisory Committee (formed in 1933) had largely come to nought, and the market mostly stabilized from the US needing to import wheat during the Dust Bowl (and the major wheat producers obliging in providing exports).
So in summation - it seems very unlikely to me that the USSR provided any substantial food aid to the United States during the Great Depression. Perhaps some nominal amount would have been provided for propaganda purposes, but even in the 1930s, the Soviet embassy in the United States was more interested in attracting skilled American workers to move to the USSR, and was more interested in attracting US corporate investment, than in providing something for free to people in the United States. What foodstuffs the USSR did put on the international market faced strong price and production competition from other countries, and also were meant to be sold for desperately needed foreign currency.
Sources
Kotkin, Stepthen. Stalin: Waiting for Hitler, 1929-1941
US Department of Agriculture. "Crop Production Historical Track Records", April 2018. Available as pdf here.
Way, Wendy. "The Wheat Crisis of the 1930s" in A New Idea Each Morning: How Food and Agriculture Came Together in One International Organisation