For the salsa: Combine the tomatoes, chile, oregano and salt to taste in a medium bowl. Set aside.
For the burritos: With a chef’s knife, separate the fleshy stalk from the green portions of each chard leaf. Trim the ends of the stalks and chop fine. You should have about 2 1/2 cups chopped stalks. Stack the leaves and slice them crosswise into 1/2-inch thick strips. You should have about 8 packed cups of sliced leaves.
Heat the oil in a Dutch oven over medium heat until shimmering. Add the onion and chard stalks and cook, stirring occasionally, until softened and lightly browned, about 8 minutes. Add the garlic and cook until fragrant, about 1 minute. Add the chard leaves (it’s best if they are still slightly damp from washing) and salt to taste. Cover and cook, stirring occasionally, until very tender, about 7 minutes. If the greens are soupy, remove the cover and simmer for a minute or two. Adjust the seasonings, adding more salt if necessary.
Lay the warmed tortillas flat on a work surface. Spoon 1/2 cup of the rice over the bottom of each tortilla. Top the rice with some chard (roughly 1/2 a cup) and 1 tablespoon sour cream. Roll the tortillas, tucking the side toward the center to form neat bundles. Slice each burrito in half and serve, passing salsa at the table.