You might be able to maintain an established food plot area with an ATV. But you would most definitely need a small tractor to establish the plot.

One another note, you could probably establish a food plot with a riding mower. It would take longer than with a tractor, and you'd have to fabricate your own implements. If you already have a welder and access to a metal scrapyard, this may be what you want to do. I've built some implements for my tractor (bucket teeth, boom pole, 3 pt sprayer). The only things you could build for a mower would be some sort of disk, harrow or other tillage device. I guess you could fab a broadcast spreader and sprayer too.