1 pound of fat = 3,500 calories
1 stone of fat = 14 x 3,500 = 49,000 calories
4 months = average 120 days
Daily calorie deficit required = 408 and a third
Weakly weight loss < 1 pound average
This level of consistent loss is very is safe and sustainable, so yes you can do it, there are some serious buts.
Don't try for quick fix dehydration diets, low carb, high protein style that will encourage your body to store more fat during and after. This is great on the scales at the end of the week, but the lost weight will soon be found.
Remember this is a long term goal and will require you to commit to it wholly. You want to be using over 400 more calories than you consume every day for 4 months, every day off pushes this number higher and when you go over 1,000 deficit you go into risk of starvation zone and that is a bad place to be.
Diet is diet, get it wrong the first 3 letters become relevant. Vegetarian, vegan or omnivore the key is always balance. Check out some food pyramids for proportion, not portion number guidance. Major on the starchy, bland stuff like pasta, rice, potato etc. next largest part of your diet is main vegetables and fruit, then protein rich, soy, quorn etc. and have oils, sugars, sauces etc. as the tiny cherry on the top. Where people get this wrong is having chips and forgetting this covers the starchy food and enough treats for most of the day, then adding sauce or a desert.
As you say many vegetarians have pretty poor diet, but a bit of reading is all it takes to correct this. Take the time and check it out.
Exercise wise, variation is best for weight loss and most other things, but if you find you enjoy some things more than others, stick at them. It is quicker overall to do stuff that is slower and you enjoy because you will keep at them rather than quit.
You are doing basic static state cardio which is a good start, mix it up with some intervals and possibly weight training if you have the facilities/desire to do so, if not let us know what you have. Running can be interval training by using the run walk system, so you could run 1/4 mile, then walk the same and keep repeating. This will soon bring your mileage up and as long as you don't stop between will be a great way to find yourself running well over a mile overall.
Final thing, when you have a level of weight loss like this it becomes easy to become despondent because you gain one week and lose another due to water fluctuation, assess on trend only not singular weeks. Even take daily measurements and trend them to get direction.
