I'm 15 years old, I'm around 9stone10lbs (134lbs) and I'm 5ft 6inches.
I'm currently trying to lose weight to reach a goal weight between 8stone7lbs and 9stone.

I've been on a consistent diet since last year and I have lost just over 2stone, although in the last 3 months I haven't had any weight loss, despite going to the gym every day and eating healthy foods in controlled amounts. My mum thinks I'm not eating enough for my body to lose weight. On an average day I eat:
Breakfast: Porridge oats (roughly 50g)
Mid-morning snack: 3 ryvita’s OR berries
Lunch: Tuna salad
Afternoon snack: Homemade vegetable soup
Dinner: Vegetables and gravy with a vegetarian chicken fillet OR salmon and vegetables

I admit that in the past few weeks I have been eating a lot less - skipping meals like breakfast and lunch - due to feeling sick and not wanting to eat. As an alternative I've mostly been nibbling on things like ryvita’s and berries throughout the day.

Any advice on any foods I'm lacking?
Foods I should stop eating?
Maybe a daily calorie intake recommendation?
(Baring in mind, I'm a vegetarian)

I didn't see anything about any physical activity. Diet alone won't help you reach your goal, you'll need to have a fairly active lifestyle.

Why are you starving yourself? You do realize, don't you, that not eating causes muscles tissue to be used as food for your body, leaving fat cells. Thus, by not eating, you are making yourself fatter. Your body is still developing, so you don't want to stunt your growth now and always have problems with physical illnesses, a weak immune system, when being pregnant, with brittle bones, and with skin and muscles.

Does your family know you are starving yourself?

I am very active, I go to the gym everyday or every two days.
I'm not starving myself, I'm still eating because I know that if I don't eat then I'll end up gaining weight and the body goes into shock and stores fat. I don't think I phrased it properly above. I am eating, just not as much as I was before. My family does know what I'm eating and not eating. My mum made sure I ate a proper dinner this evening and I ate it. What I was trying to say above was that I don't really have much apetite recently.

You said, "skipping meals like breakfast and lunch." That concerns me very much.

Who said you need to lose weight?

Well I had been skipping meals; I hadn't eaten proper meals like usual. More of small snacks if that makes more sense?

My family and friends say that I don't need to, but I have a 'belly' which I'm self conscious of and I don't like some fat on my legs so I want to get down to 9 stone just so I slim out a bit more.

But the weight may not come off where you want it to. I think you are doing this wrong.