Answer: Yes, an immoral person can in fact be happy.
Explanation: Being immoral is simply a lifestyle, there simply is no answer to WHY a person should be 'moral' in the first place. You can certainly be happy and be a non-ethical person. Morality is literally pertaining to one's purpose and beliefs in life, we can view ethical behavior as a pathway to achieve a moral life; hence, a happy life. Simply, people who are immoral have come to terms with being immoral, they can in fact be happy.