In a large skillet, stir the oregano over medium until fragrant, about 15 seconds. Transfer to a bowl.
Heat the lard in the skillet until shimmering. Add the garlic; cook until golden, about 2 minutes. Add the onion; season with salt and cook until softened, about 2 minutes. Add the cream; reduce the heat to low. Simmer, stirring often, until thick enough to coat a spoon, about 30 minutes. Add the shishitos and oregano; simmer until peppers soften, about 10 minutes.
Place 3 tortillas on each plate. Top with the shishito mixture (it will be very saucy). Squeeze a few lime wedges over the tacos; serve the remaining wedges on the side.