Yes, of course. The USSR was starved for western money, and would actively recruit tourists. It was expensive, and you didn't have any freedom to just, say, go into the countryside on a whim, and the KGB would follow you around, and if you were in the country when a U2 got shot down you would get interrogated, but you could do it, certainly.

Their travel agency was called Intourist. Robert Heinlein wrote a review of it, including the experiences described above, in Expanded Universe, a nonfiction posthumous book. He also writes about the USSR in Grumbles from the Grave, another posthumous nonfiction book.

There were other ways of visiting as well. If you were a "fellow traveller" socialist, they'd sometimes arrange tours of the country, especially in Stalin's time, and you can go there via other official programs, such as a Fullbright Scholarship, which a colleague of mine (Richard Jensen) did in the 80s.

Typically* you would have to be invited by the Soviet Union to be allowed to visit. This would mean applying through the embassy. I do not know of anyone going to Russia for a vacation, however during that time. The majority of the travel between East/West was exchange students. Soviets would be allowed to attend certain universities and vice versa. My adviser in Undergrad was one the ones that traveled to the Soviet Union through an exchange program.