Well, theoretically you can. But you have to be a really high on the ladder.
Brezhnev had several good _imported_ cars. Gagarin had a sport car (gift from French, I believe). Vysotskiy (think of a Soviet Elvis Presley) had a good car.
But other 99.99%... no chance.
Of course, I'm not taking into accounts hundreds of "expropriated" German cars which appeared in USSR after 1945.
Can you imagine John Glenn being presented with some underwear in the White House?
As for spending billions on vanity project, the manned flights have reasons other than political. Check here - http://www.lettersofnote.com/2012/08/why-explore-space.html .
There certainly are reasons for space exploration. In a country where a year after Gagarin's flight army shot unarmed civilians who were demonstrating asking government to give them some food, it is far harder to justify.
The other problem being, of course, that whether you were given $100K, or a million, you couldn't really go buy a car or a house anyway. And, apparently, underwear or a decent coat either.
https://en.wikipedia.org/wiki/Moskvitch_2140
https://en.wikipedia.org/wiki/Lada_Riva
https://en.wikipedia.org/wiki/GAZ-24
There were a few models concurrently of the two popular brands, but they were not great.