Without knowing how tall you are, it's impossible to say whether you are overweight or not. If you are my height (I'm 5'6", which is a bit over 167 cm), then you'd be underweight. If you are 122 cm (about 4 feet tall, which is rather short if you've finished growing), you're overweight. It's rather difficult to get your weight to come out as obese, seeing as, in English units, you're around 88 pounds. Which is tiny, very thin, unless you are extremely short -- short enough that you are either not finished growing or are genetically a dwarf or little person.
So let's work with that -- you are, most likely, not fat by any stretch of the imagination. You're, more likely, just not toned.
So you need to do some exercise, if you feel like you're jiggly. That's fine, but recognize that you need to fuel yourself in order to do that. That means eating, and eating good food, and keeping it down. Good food is what? Chicken, fish, salads, vegetables, fruits, dairy products. Exercise? Things like walking, swimming, jogging if you are up for it, and possibly even a bit of weight training. Women have to do an enormous amount of weight training in order to really bulk up -- don't worry about becoming huge. You most likely won't. It's a lot more likely that you'd just become firmer.
And, as for the idea of throwing up in order to achieve weight loss, or to do anything other than as part of a disease process, it's not exactly a glamorous or wonderful or sound way to achieve your goals. Still feeling like it is? Then I suggest talking to your doctor about it. You're very young, and you can do a lot of damage to yourself by throwing up all the time.
I've got no problem if you want to eat better and become fitter. Those are great goals for anyone, at any age. But I'm about 100% convinced that you don't need to actually lose any kilos. And no one here will give you lessons in how to better throw up, okay?