Can I leave the okara in when making tofu?

The okara has some unextracted protein and sugar but it also has a large quantity of fiber. That fiber will prevent the protein from properly coagulating. If you leave in all the okara you get soybean porridge.

I imagine you could leave in a portion of the okara and you would get a fragile but hearty tofu. The problem with this is that, with most tofu recipes, the beans are not cooked long enough for the okara to be palatable. With all that sugar it would also give rise to much more flatulence.

You don't have to throw the okara out. There are many recipes that make use of the okara for its fiber. It is pretty straight forward to use it in baked goods. Toasting it gives it more interesting flavor.

If you want to look into tofu-like products that use the entire bean you could research tempeh. The whole beans are fermented so you get a meaty product with all the nutrition of the whole soy bean.

If you include the actual beans in the curds, it'll be edible, but it won't really be tofu.

(Edited to add: There are a couple of Japanese companies that produce a product called "Tofu with okara", but based on my reading of this article plus the help of awkward machine translation, it appears to be using okara treated with an enzyme in order to reduce the amount of waste product. Probably not the sort of thing you'll be able to replicate at home, but worth noting that you aren't the only one with this concern; okara consumption, and use in animal feed, has been declining in proportion to the level of consumption of tofu itself in Japan.)

Instead of throwing it away, why not use the okara in other dishes instead? I like to mix them with vegetables and seasoning (soy sauce, mirin, salt, or western seasonings if preferred) and deep fry them as "croquettes", which I serve with a tonkatsu sauce or similar. I've also made an okara burger from time to time.

I've also mixed them with some mushrooms, onions, carrots, celery and some vaguely Italian seasonings, deep fried them, then served them in tomato sauce. In my experience, they tasted nice the first day but didn't hold up as leftovers.

Okara can also be simmered with carrots, green vegetables, and perhaps some hijiki along with mirin, soy sauce and sugar and used as a side dish.

You can bake the okara at a low temperature in an oven to dry it out a bit, then use it in things like cookies. (May only appeal to Japanese sensibilities but they were popular in Japan for a few years).

I just mixed my latest batch of okara with an equal amount of diced chicken, some finely minced onion, peppers, and a bunch of spices. It made some very nice "burgers", the leftover ones likely to go into my current, ongoing batch of India style curry. Been doing this since discovering I have elderly lactose intolerance, and make soymilk regularly. Tempeh and soygurt are on the list, too.

The other answers explain why you don't leave them inside the tofu. Indeed, if you leave them in, the tofu won't coagulate; however, if you allow the okara to ferment, then you get something resembling fermented tofu.

In China, we often use okara/douzha to feed fish, birds, etc. (if you have one). We also use it to make liangban (cold-stir, "salad") dishes, so you might try adding it to salads.

You can also make "meatballs", or add it to pastry, etc. The opportunities are endless. I've also seen people use it as a crust for deep fried dishes!