Hi OP
It sounds as though you have a sensitive and thoughtful little boy on your hands, which is a great thing :)
I know it's a worry with a child who doesn't have a great appetite or who is fussy, but I do think you should respect your DS's wishes on this or you could damage his relationship with food.
I've been vegetarian since I was 10yo - the rest of my family weren't veggie. I have two DC aged 9 and 6 who have been raised on a completely vegetarian diet. They were born large and healthy (both over 9lbs even though I'm fairly slim) and still thriving even though they're both a bit fussy about food.
It's perfectly possible for your DS to have a nutritious vegetarian diet. Does he like eggs and cheese? Baked beans? Would he eat a 'fake meat' product like a veggie sausage or burger?
Things like lentils and beans can be easily blended into soups and pasta sauces.
If he's underweight there are lots of high-calorie veggie foods - full fat Greek yogurt, grated cheese on everything, buttery toast, peanut butter, avocado, olive oil, etc.
Just make sure you're offering him something with protein a few times a day and the same with iron. As much variety as possible with types of protein and types of fruit and veg.
Breakfast cereals have added iron, which is absorbed better with a glass of orange juice on the side. Dried apricots are also high in iron. So it's not just leafy greens...
Good luck!