A "vice" in this case means a weakness. It is often used for something that might be considered wrong in some way, like smoking or drinking alcohol. The dictionary defines it as: bad or immoral behavior or habits a moral flaw or weakness a minor bad habit http://www.merriam-webster.com/dictionary/vice

It means that it is his addiction. However, since it is only coffee then this is not such a bad thing! It is a physical defect or weakness for something. Coffee, chocolate etc. Or an undesirable habit, such as smoking etc. Something that a person feels they need to cope during the day.