If you have a grow light, you can grow quite a variety.
How warm and steamy do you want to keep an unpainted interior of your house? Seriously, Global warming is No Problem to Veggies, but is a Big problem for Paint. Most Veggies with very few exceptions [cucumbers, gourds, squashes, rhubarb, some celeries, cabbage, eggplant and pumpkins] like it 80°F and Steamy. Paint and drywall however likes it 72° and dry. I think deep down this is why we eat Veggies. The child inside we says: 'we don't like veggies, but if we don't eat them, who will? They might eventually evolve and want a Sauna we could never bear 24/7'. Trust me...I just got done with 5 months of a broken Furnace, nothing but steam and hot water in my 7200 cubic feet Doomsday Prepper Den Underground. Had to bring up all my Canned Goods to prevent rusty cans. Great environment for Tomatoes, Bananas, Tangerines, Mold, Mildew, Maggots and Moss. Bad environment to dry laundry. Peeled all the Paint off my drywall in the stairwell. Rotted some of my wood. Attracted Fleas from the Dog next door. Had to Bomb twice monthly. Killed my freezer and I not only lost $350 in food, I also had to deal with Maggots and Flies too. Yada yada yada. Nearly puked. Good God am I glad that ordeal is over! Basically my advice is a tad bit biased, due to 5 months of mental torments. But I do recommend that you instead of considering Veggies, instead Host a Herb Garden, Berries and Dwarf Fruit Trees that like Zone 5 to 7. Why? Plaster, Drywall and Paint like Zone 5 to 7 more than most herbs and trees do. Anything that Grows above 45° but beneath 76° is great indoors. A Great place to get these is at: Fastgrowingtrees.com because they filter out by a search any of the junk that dies indoors. They have mostly everything: Apples, Grapes, Pears, Peaches, Mangoes and mostly anything you could ever desire to eat fresh. Good Luck!
I agree, as long as the plants will get enough light from a grow light, root veggies like carrots, etc. will probably be the only ones that would be challenging because you need to plant so many to get enough to eat. A sunny window will not be enough light for them to grow healthy and produce a lot of veggies.
I think you'd need to stick to quick croppers such as salad leaves and herbs if you want to grow veg indoors. You could also try mushrooms if you have a dark shed or similar :)
Sprouts are great options