I am going to assume they are not asking you to lose weight that would leave you below a “normal” weight for you. Meaning, they are not asking you to be dangerously thin.
What I would suggest is to count calories. It really is about calories in the end. It is a math problem. If you figure out the weight you want to be, thereis a specific amount of calories that lead to that weight, regardless of your height, etc. I can dig up the calculation if you want. Your other choice is to figure out how many calories you eat, and reduce what you are eating. Approximately 3500 calories a week is 1 pound. So, if you eat 500 calories less a day, you will lose 1 pound a week. Although, the first week you would likely lose 2 pounds because just by cuttng food you reduce sodium, so you will lose water. If you cut back a little on salty foods, you will lose another few pounds of water. BUT, salts are important for your body, for your heart, so do not do anything too extreme.
Being vegetarian certainly does not mean you eat healthy or low calorie necessaily. I will assume you are lacto-ovo since you did not write vegan, and dairy foods have a ton of fat and calories. Obviously pastas, and other starches are high calorie also.
You said you did not want to add more exercising, but maybe do little things like parking farther away from the entrance door of the supermarket and work, and climb stairs rather than using elevators.