So I guess there's 3 parts to this question:
I see a lot of headlines that do say she "bought" a tank, but that's not an accurate description of what she actually did. She sold her personal possessions and raised extra funds to donate to the Soviet war effort, and wrote a letter to Stalin requesting that the funds be used to 1) build a T-34, and that 2) she be trained and given command of that T-34. Stalin (no doubt seeing the great propaganda opportunity) said OK. She was given five months of tank training in Omsk and then the requested T-34 was delivered to her unit from the production facilities.
So it definitely wasn't a case of her going out and privately "buying" a tank, any more than when other countries had donation campaigns to raise funds for, say, constructing a battleship.
Now for the real answer: the official story says that she has collected 50 000 roubles (mostly her and her husband's life's savings), donated it to the state war fund and then sent a telegram to Stalin asking to be allowed to fight as a tank's driver. Ostensibly Stalin himself ordered that she would be allowed to do so. She was featured in propaganda, received a full five-month course in tank driving and maintenance (when most of the conscripts had significantly shortened ones) and finally sent to the front.
50 000 roubles is also not enough to pay for a tank. Even ignoring the peculiarities of war economy the actual cost of T-34 was at least 130 000 (and up to 274 000) roubles.
So the actual sequence of events (ignoring possibilities of it being fabricated): a person makes a huge donation to the state war fund and asks to serve in specific capacity. Her request is granted. If you were a person with significant amount of money on hand, heard about Oktyabrskaya's story and wanted to do likewise - you would be free to try but there would be no guarantees at all - it is possible that your donation would be accepted and your request ignored.