The honest truth is, it can be as expensive as you want to make it. If you feel that, eating vegan, it is necessary to have some kind of "meat" on your plate and you spend a lot of money on the imitation products which are pricy, your grocery bill won't go down any.
Vegan is not synonymous with organic, either. Usually the two go hand-in-hand because lots of people eat a vegan diet for health reasons. But it's entirely possible to eat vegan without eating anything organic.
The key to getting all your nutrients in ANY diet is to just eat a variety of foods. Beans, nuts, greens, veggies, fruits, whole grains. A variety of those each day and you're good.
It's easy to turn any diet into an unhealthy one. If you're a meateater and all you eat all day is meat and potato chips, you'll have nutrient deficiencies just as easily as a vegan who chooses to eat just black beans and vegan cookies day in and day out. That's why the modern "American" diet which is heavy in fried foods, processed foods, meat and other animal products is so dangerous. They've left out all the other kinds of foods (beans, nuts, greens, veggies, fruits, whole grains) for the sake of all of this other junk.
And to answer the question, "all that other junk" might be cheaper on your grocery receipt, but what do you get for your money? No nutrients, no useful calories, nothing really, except a higher risk of dying earlier. If you replace the junk with healthy food your bill might go up, BUT you'll actually be getting what your body needs in return instead of just, well, garbage.