Yes, the Soviet Union had a tax system. In broad strokes it worked in a pretty familiar way, but it was also different than capitalist systems in the same period. I'm not able to dig up a lot of really precise numbers on things like income tax rates off hand here, but I can talk about the system and the kinds of taxes there were.

First off, I'm going to assume that you're not talking as much about the grain requisitioning that took place especially during the Civil War and during the 1930s and that resulted in famine. That is an important topic and definitely has something to do with taxation, but it's almost it's own little sub-topic. The short version is that the Soviet Union imposed incredibly harsh grain requisitioning on the peasants in the countryside in order to both feed urban populations and to sell internationally as a way of raising money for the mass industrialization that was underway at the same time. This resulted in devastation in the countryside, including mass famine when combined with a couple of other factors. I'll spend the rest of the time talking about more conventional taxes.

The most substantial tax was the turnover tax, which functioned in a very similar way to a V.A.T. There were a variety of other taxes, including income taxes, which provided a relatively low percentage of the total state income. For example, in the Soviet Union in the 1950s, income taxes accounted for only about 10% of the total income, while the US government in the same period was funded about 50% by income taxes. Note that this isn't a commentary on marginal income tax rates, but rather the structure of the entire system. Of course we have plenty of countries with a VAT these days, but the Soviet structure of a centralized economy made extracting taxes at places other than the final sale of goods generally quite appealing and "easy."

HOLZMAN, FRANKLYN D. "INCOME TAXATION IN THE SOVIET UNION: A COMPARATIVE VIEW." National Tax Journal 11, no. 2 (1958): 99-113.

The Soviet Union also imposed taxes to try to influence behavior. A good example of this was Soviet post-war pro-natal policy that taxes families for being childless. The Soviet Union lost somewhere in the neighborhood of 25 million people in World War 2. After the war there was a great deal of concern about population and population growth. The tax on childlessness was part of the plan to make having children more appealing. This kind of tax is common in other places and times as well.

For more on pro-natalism under Stalin:

Hoffmann, David L. "Mothers in the Motherland: Stalinist Pronatalism in Its Pan-European Context." Journal of Social History 34, no. 1 (2000): 35-54.

It's important to note, however, that at the level of the individual, net income and cost of goods, the things primarily effected by income tax and VAT were actually far less important than they were in capitalist economies. This is because the Soviet economy was in many ways defined by a shortage of consumer goods. Money is (obviously) only useful when there are things to actually buy. Unemployment was low and the distribution of wealth in the Soviet Union was comparatively low when compared to the United States during the same time period, but the way to get your hands on goods had a lot less to do with cash on hand than it did having good social connections. There has been quite a bit of work on this topic. The Russian name for this informal economy/system was "blat" and a great deal of getting by in the Soviet Union had to do with managing connections outside the official economy system.

For more on that topic see Alena Lebedeva. Russia's Economy of Favours: Blat, Networking and Informal Exchange. Cambridge University Press, 1998.