If I don't eat fat, and eat only other things, for example protein and vitamins, then does this reduce my chances of getting fat? In other words, I will never be fat even if I don't do exercise?
It is possible that if you eat nothing but protein, you may starve even if you're theoretically eating enough calories. The reason being that there is a maximum amount of protein your liver can process for calories per day, and it is below the number of calories you burn per day. This is called Rabbit Starvation Syndrome (after settlers who were starving despite plenty of lean rabbit meat), though it has not been well studied in humans. In any case, that's probably a pretty unhealthy way to diet, since it would put a lot of stress on your liver (to turn protein into glucose and urea) and kidneys (to clear the resulting urea).
To answer what I think is the intent of your question: within reasonable limits, your body is very efficient at converting calories from all sources (protein, carbs, alcohol) into fat. So yes, you can become fat as long as you are eating more calories (from any sources) than you expend, and are not eating a radical, unhealthy diet like getting nearly all your calories from protein. Marion Nestle goes into much, much more detail in her book "Why Calories Count", which is probably worth reading if you're interested in losing weight.
Only proteins or vitamins?No carbohydrates and no fat?? Well the thing is, if you eat more than your body burns, you will become fat, even if you don't eat fat at all. Carbohydrates are very energy dense, so if you eat too much carbohydrates your body will store the extra energy in fat cells. So if your daily energy intake is too big you'll get fat.
Also the thing is I don't believe it's very healthy just eating protein and vitamins as you said, you really should have a balanced eating routine, with fat, carbohydrates, proteins, vitamins and so on. Even if you don't exercise you won't become fat if you don't eat too much. There are plenty balanced eating plans out there, just google it and I'm sure you'll find something that's healthy and makes sure you won't become fat.
Hope that helped.