How can a lacto-vegetarian on restricted diet (no salt, sugar, oil, vinegar, gluten) eat healthy?

Balance the diet. A lacto-vegan has many choices for foods. A good nutrition book or a visit to a dietician can help form a balanced diet. Eating a big variety of foods helps prevent nutrient deficiencies. Also, one can take a daily multivitamin+mineral tablet. Lowfat or nonfat dairy products have plenty of protein. One can buy natural, gluten-free foods and cook them without adding salt, sugar, oil, nor vinegar.
See below. Since you are a lacto vegetarian i would suggest to stay away from regular milk as it contains casein which some people are sensitive to and it may lead to inflammation. Make sure you also get your fruits/vegetables (eat what is in season, try to eat organic). Protein can also come from lentils/beans/tofu, etc. Good luck!