Cook linguine in large pot of boiling salted water until tender but still firm to bite, stirring occasionally. Drain, reserving 1 cup pasta cooking liquid.
Meanwhile, cook bacon in heavy large pot over medium heat until beginning to crisp, about 10 minutes. Transfer to paper towels to drain.
Drain all but 2 tablespoons bacon drippings from skillet. Add onion and saute over medium-high heat until softened, about 7 minutes. Add Swiss chard and sprinkle with salt and pepper. Add pasta cooking liquid to skillet. Toss until chard is wilted and tender, about 4 minutes. Sprinkle vinegar over; cook 1 minute.
Add linguine and oil to sauce in pot and toss to coat. Transfer to large bowl. Sprinkle with bacon and cheese. Season to taste with salt and pepper.