Both "I feel like coffee" and "I feel like having coffee" are correct. "I feel like..." can be followed by either a noun or a gerund.

Some people will shorten their sentences for the sake of brevity. I feel like having is the way we would write the sentence, surely. In speech though, people will adopt the shorter form sometimes. "I fee like hot food today." "I feel like one of those Latte' coffee drinks today." It seems a little unusual to see a person compared to some object or type of thing this way, I know.