I'm not getting enough calories. Should I force myself to try and eat more?

I'm a 5ft 3, 119 lbs and 16 years old girl. I started eating healthier and counted calories just out of curiousity. I found that on average, I get around 900-1000 calories per day but I heard I should be getting a minimum of 1200. I don't try to starve myself or anything, in fact I eat quite a lot (six times a day) but I eat healthy foods. I eat a lot of protein (mainly from eggs, protein shakes and beans), carbohydrates (whole grain bread, brown rice, whole grain cereal), low fat dairy products and a lot of fruits and vegetables-not all in one day of course. I do try and make it all balanced and eat when I'm hungry. Still though, I only manage to get 900 and feel full. Is it because I'm vegetarian? Should I make myself eat more or does it not matter if I feel full with only 900? Would I put on weight only by eating that many?