Traditional gastronomy of Ecuador offers some vegetarian options. Just keep in mind that dairy and eggs are quite common.
Ecuador has 4 different climatic zones and it makes 3 different regional cuisines: Costa (ocean coast), Sierra (andean mountains) and Oriente (Amazon jungle). This article is more about andean food.
Street food is very popular and it's easy to find something vegan. You'll see lots of vendors selling corn and fried plantains, just ask it without cheese. Plantain and potato crisps with tomato sauces are also available. But the best choice is ceviche de chochos. Normally, ceviche is a dish made from fresh fish, which is typical to many latinoamerican countries and originated from Peru, but in sierra of Ecuador it's made from a local kind of beans - chochos or tarwi, which botanic name is Lupinus mutabilis. This bean has even more protein than soya (up to 50%).
In any local market you will find lots of fresh local fuits and vegetables. Ecuador produces some quinoa and chia, so you can buy it for cheap. Some great ecuatorian recipes could be found here, just veganize some of them replacing milk.
Although vegetarianism and veganism are not so common, you could easily find a vegetarian restaurant even in small cities. In any local market you can have a fresh made juice or a fruit salad. Ecuador has its nice special: come y bebe, which is fruit salad with juice. Remember, that normally fruit salads come with milk, so just ask to make it without it.
If you can't find any suitable vegetarian option, you can ask in any casual restaurant to serve you food without meat. Actually, you will have to clearly tell them what you don't eat, so spanish conversational skills are required. Usually you will have a typical latinoamerican rice'n'beans food. Some spanish words that will help you: Arroz - rice. You will eat your pasta somewhere else. Menestra - bean or bean dish. Think of lentils with some spices (a bit). Ensalada - salad. Ensalada de frutas is a fruit salad. Sin carne, leche, huevo - without meat, milk, egg.