This is sort of a weird question. It depends what you mean by happy. There is the kind of happiness that means feeling mostly OK and satisfied with your lot and not feeling miserable and there are those moments when you just lose yourself and feel at one with the universe. I think everyone feels that latter type but alas very rarely. I don't think it's got much to do with your physical self or your appearance or even your circumstances. It's a sort of gift. Of course it all depends on your psychological makeup and many things besides.
I don't believe it's impossible to be happy if your fat but the overall feeling of wellbeing is unlikely to be that great for most people. Also, it's very hard to dissociate your own feelings of self-worth from the images that society values or does not value.
Personally, for what it's worth, I feel that being overweight most of my adult life did have an adverse impact on my feeling of wellbeing, physical and mental. However, if I'd been slim I might not have been any happier. Actually I manged to be reasonably happy, all told. I went out in the world and did stuff and met people and pursued my career and had a host of experiences, some good some bad.
I don't know whether I've answered your question but I've tried. :-)