I'm reading Anthony Beevor's book "Berlin" and he mentions the following:
"....Komsomol members also helped the barely literate ones write letters home. They were particularly proud of having bought a T-34 tank with their own money. Their tank ‘Komsomolets’ had already ‘destroyed a few enemy tanks and other armoured vehicles and crushed many Fritzes under its tracks’."
Could Russian soldiers buy a tank themselves? How would this work?
You could buy a tank, but that does not exactly mean you pay money in a store and you get a tank in return. Buying a tank meant that a person (or, more likely, an organization) donated some amount of money, perhaps large enough to buy one tank, perhaps enough to equip an entire tank unit. Except for a few cases (for example, Mariya Oktyabrskaya) the person paying for the tank would not actually drive in the tank. What they would get to do was pick an inscription and/or an insignia to be written on the tanks.
I have data on the amount of money donated to the Red Army to build tanks here and a record of some of the tanks that were paid for by civilians here
There is also a humorous piece written by Mikhail Zhvanetskiy called "My Armour" (Bronya Moya) about "buying a tank with donations from artists, like during the war, but to use it myself for some time" along with "a friend in a helicopter to fly ahead of me and a few friends with assault rifles to run right behind me".