You don't need work to be happy, but having a job that can pay the bills certainly helps a lot. You will more likely be unhappy if you don't have one. I need 20 more wotds, so this will fill in the rest of my post. Thank you for your time.
If by "work", we mean that a person must engage in labor for the purpose of physical subsistence (whether physical or mental; directly producing one's own food or working for income to exchange for supplies...) , then no.
But if "work" is defined as using one's time in a "productive" manner, then I would suggest the answer is yes. At least for me it certainly is. If left entirely to my own devices without any concern for income or subsistence, I would still spend time doing things that qualify as "work" for someone somewhere. Reading. Writing. Thinking. Playing sports. These are all things that someone somewhere gets paid to do, and as such, they are things that can qualify as "work" in some sense.
I believe most people need to feel some sort of purpose in their existence to be happy. This does not mean work. However, I do believe people need to feel like they have a reason to live. This could be done through numerous other factors like parenting or being someone's friend.
Work does not really guarantee us happiness. There are a lot of things that could make us happy. For example, going out with family or doing something that you love like teaching, cooking etc. Work only comes as a source of happiness when all other things that you do does not gives you anything but tiredness etc.