Yes, but only if they're looking for investors. You would need to contact them directly. Unless you're looking to invest a significant sum, they may not be interested in speaking with you. (Think at least 6 figures, maybe 7 depending on their size and needs). This is otherwise known as being a Venture Capitalist. Some companies don't want additional investors because the capital isn't yet needed and they don't want to give up shares in the profit/control.
Alternatively, you could try and figure out which investment groups already have a stake in the company you're interested in. If those companies are publicly traded, you could buy stocks for their company with the expectation that their stock price will increase if the company you know of does well in the long run.