To buy a ticket for the Vilnius-Minsk / Minsk-Vilnius train you will only need your passport data: full name and passport number and, naturally, a card to make an online payment. The system will generate an electronic ticket which you shall present at boarding.
If you are short of time or the website looks scary or your US card isn't accepted by the website of the railways, I can buy you tickets for either Vilinus-Minsk train or Minsk-Vilnius from my end. I will only need the data mentioned above and your email. Please contact me for details.