There are some kernels of truth to this story for the Russian Imperial Army during ww1 and some for the Narodnoe Opolcheniye (people's militia) units of the Soviet army during ww2.

The Russian Imperial Army was unable to equip all its drafted men with rifles early 1915 - and purchases and deliveries of British and Japanese rifles were made (remaining rifles in stock were later sold at high prices to the desperate Spanish Republic 1936-38). By December 1914, the Russian Imperial Army had roughly 6,5 million men in uniform, but only 4,5 million rifles to equip them with. At the end of 1915, they had 2 million men at the front, but only about 1,2 million rifles to equip them with. However, by that time the Russian production had picked up, reaching about 1,3 million rifles per year, and there were 2 million rifles in Russian ports or on their way from Britain and Japan.

All countries fighting in ww1 suffered huge problems trying to supply their forces and equip men raised to replace the extreme casualties suffered by the pre-war armies during the first year of the war. It took time for industry to change to war production, and the lack of nitrates (in the Central Powers due to the blockade, in the Entente due to difficulties in increasing shipment of guano) hampered production of ammunition. The British had to purchase Canadian and US rifles and took rifles from the Indian army as a stop-gap measure. However, the Russian Imperial Army was the only army to send uniformed but unarmed men into combat, expecting them to arm themselves on the battlefield late 1914 to late 1915.

During ww2, the Soviets attempted several times to use Narodnoe Opolcheniye, or people's militia units as a desperate stopgap measure when the Germans advanced deep into their territory. These were civilians organised in a sort of a home guard and at times trained decently well and equipped almost like regular troops and at other times thrown into combat barely equipped at all when there was little or no time for training and little equipment available. This was especially problematic during Autumn 1941, when much of the Soviet war industry was being moved from territory that had been or would be over-run to beyond the Ural mountains, causing a temporary slump in supply and new equipment available to the Soviets. For example, the German 6. Panzer-division, when advancing on Stalingrad as part of Fall Blau in summer 1942, encountered a Soviet unit of AA guns and attacked it, noting that the enemy was firing badly, and after running over the unit, discovered that the guns had been manned by female factory workers still in civilian clothing. On the other hand, the Narodnoe Opolcheniye units part of the 62. and 64. Army that defended Stalingrad seem to have performed well during the fighting.

It is quite possible that Narodnoe Opolcheniye units were at times equipped with a lower number of rifles than it had men and went into battle like that - they often lacked heavier support arms such as artillery, AT guns and mortars. However, I have not seen any records of systematic under-equipment of Soviet units during ww2, neither regular forces or Narodnoe Opolcheniye.

Sources:

Brusilov Offensive, by Timothy Dowling.

Race to the Front, by Kevin D Stubbs.

Stalingrad, by Anthony Beevor.

2194 days of war, by Cesare Salmaggi and Alfredo Pallavisini.