I used to live a bit south of there. Richton Park never has had much....doubt they do now. The only places that used to be there were one adequate Mexican place, an Aurelios that was take-out only, and Chicago Pizza Company. For all I know, all are now gone.
Fortunately, a short drive north and there are many more options. Rt 50 is probably the best way to find them.
The first town north is Matteson. There used to be some chain options which I think still exist, from Red Lobster (not probably very helpful), to Olive Garden to IHOP to Cracker Barrel. That's mostly to the west. To the east of Rt 50 is Ciao, a nice Italian place.
The next burb to the north is Olympia Fields. Given that my SIL just ate there last week and liked it, Burgundy Bistro may be a good option. They have a gluten-free menu, which doesn't help you, but they have recently re-opened and I've also heard good things from a group of celiacs who went there for dinner two weeks ago. There is a Szechuan restaurant just a couple door away from Burgundy Bistro that I've not been to in a long time, but was always decent. I think my SIL likes it.
Next is Flossmoor; in fact, some of the following restaurants are very close to the Olympia Fields ones. Siam-Thai is in Flossmoor, not too far from Burgundy Bistro. Not a clue as to how good it is. They used to be very good before they moved to this location. The Waffle House is close by, too. Besides your typical breakfast items that would be vegetarian, not sure how you'd fare.
A little farther north and a bit to the east, Fresh Starts has always been a lovely spot; I haven't heard anything bad about it. The menu changes regularly; I'd imagine they'd have several options for veggie only. Across the street is Flossmoor Station. Adequate at most, at least the last few times I was there. It's a brew pub - not sure you'd find much, but you could find their menu online and look.
Homewood, the next suburb north, has the most options, I'd think. The original Aurelio's Pizza location is close to downtown Homewood. I'm not a huge fan, but many people love the cracker crust. I think my SIL likes Mama and Me Pizza, but I'm not positive. I think Balagio is open for lunch. It's a nice atmosphere, typical Italian. I'm not a fan of most Italian restuarants, but it's always been busy. There is a sushi place on Ridge, the little downtown main street, but I can't think of the name of it. When I meet a good friend who's in Homewood, we often go there. I just looked at a restaurant guide and it appears to be Little Tokyo. They'd have veggie rolls in several varieties, I imagine. It's always been perfectly fine. Not too far away from the downtown area is Blueberry Hill Pancake House.
On Dixie Highway, there is Cilantro, a Mexican place. Again, haven't eaten there is many years. It used to be fine.
If you get to Homewood, treat yourself to ice cream at Mitchells after lunch, given that you eat dairy (and potentially eggs). It closed briefly, but reopened. It's a small, old fashioned ice cream shoppe. Fun treat.
I know I wasn't much help regarding how good these places are. I am in the area occasionally to see family and friends so have been to a few of these off and on, but not regularly enough to tell you what is the best option. At least you have some names and can look at menus, if they have them online. Enjoy your lunch.