Take it from someone who's spent the past two years trying to scrape a decent vegetarian meal out of this barbecue and steak kinda town - it may take some work, but you do have options. Here's where I make it easy for you:
(If you've searched for restaurants in Columbia, you've probably come across the following: Mr. Friendly's, Motor Supply Company, Garibaldi's, Terra, Dianne's on Divine, etc. Don't bother. Some have one veg. entree; some have none. It's not worth gearing up for a good meal, only to be disappointed.)
Hunter Gatherer: To my knowledge, the only brewpub in Columbia - has a menu really well-balanced between veg. and meat options (and their pork comes from Caw Caw Creek), a great atmosphere, great beer.
Blue Cactus: Nestled in a corner in Five Points, has probably the most extensive veg. menu in Columbia, mostly Korean influenced. The service is slow, but this is a family-run place - you can see them back there cooking out of a crock-pot - what do you expect?
El Burrito: Awesome, awesome, fresh simple taco-stand-esq. joint in Five Points. They offer tacos and burritos, and that's about it. Beans and rice and all are made without meat products. Fantastic back patio with live bluegrass on Thursday nights in the summer.
Cafe Strudel: If you want brunch, look no further. Delicious "Hangover Hashbrowns" and great atmosphere.
Delhi Palace: The Broad River Rd/Bush River Rd area of Columbia sports a number of Indian restaurants (including the most awesome Indian Grocery on Bush River Rd.), but the best is Delhi Palace, recently relocated to St. Andrew's Road. They have a fantastic lunch buffet. (But, to be fair, Punjabi Dhaba, on Bush River Rd., has an equally bangin' Sunday buffet. Just sayin'.)
Here's a good hierarchy: Pinckney's Produce CSA (in the summer), Rosewood Market, Peaceful Villa (in West Columbia), All-Local Farmer's Market, Bi-Lo, Publix, Earth Fare, Fresh Market.
A couple good CSAs have sprung up recently - Pinckney's Produce isn't located in Columbia but near enough that I think it counts, and the produce is DIVINE.
Rosewood Market is a really wonderful food store with generally nice produce, locally sourced eggs, meat, and dairy, and a great little lunch counter.
Peaceful Villa is a new storefront run by family farmers - their produce is lovely and completely chemical-free.
The All-Local Farmer's Market is far preferable to the State Farmer's Market (which is weird and, I think, currently in transition to its new location) - it's small, but I generally buy fabulous eggs from Wilmore Farms and good bread from Heather's Artisan Bakery. Each week usually holds a surprise (fresh spinach pasta one Saturday, strawberry donuts another).
Surprisingly, Bi-Lo has been sourcing some nice produce these days, and it's often local.
Publix's produce is a last-resort, since I often find myself buying peppers from Holland when I shop there.
I stay far away from overpriced Earth Fare and/or Fresh Market, but it can come through in a pinch. (Although Earth Fare didn't have dried chili peppers the last time I went... for real? What self-respecting health food store doesn't carry dried chili peppers. FAIL.)
OK, so those food establishments listed above really get me through the days and weeks here. It's no NY or San Francisco or even Raleigh-Durham (where I used to live... oh, the luxury), but if you hunt hard enough you can eat veg. even in Columbia.
You just might have to cook at home a lot more.
(Oh, and forget about decent falafel. Pitas and the Mediterranean Tea Room are dreadful.)