Yes, of course they can. States throw people in prison all the time.
As someone who's not from the states, this question is understandable. But a federal offence is one that, typically at least, affects the entire country, or more than one state.
Way more people are in state prisons than federal prisons, because most crimes by volume are state crimes. If I were to cross the street and shoot my neighbor's dog, I'm going to state prison. If I go even further and rob the gas station while I try to flee, state prison. If I get to Montana and kill someone, that's still not a federal crime, and Montana and New York get to argue who gets me. By contrast, if I set up a massive fraud scheme and take millions from people in more than one state, that's federal.
There are some exceptions, things which the government finds illegal but states don't. But each state has their own crimes, a lot of them overlap, and yes, they can and do sentence people with fines, prison, or even death.
Hope that helps.