Combine masarepa, cheese, water, and vegetable oil in a medium bowl and knead with hands until a dough is formed. Take a small amount and flatten it between your palms. If the edges crack, knead in more water, a tablespoon at a time, until dough is supple and smooth but not sticky. Season dough to taste with salt.
Cover and set aside for 5 minutes.
Divide dough into 4 even pieces and roll into balls.
Flatten each ball down to a disk about 5 inches in diameter and 1/4-inch thick.
Melt butter in a 12-inch cast iron or non-stick skillet over medium-low heat.
Add arepas and cook, moving them around the pan and rotating them occasionally, until first side is charred in spots and a dry crust has formed, about 5 minutes. Flip arepas and cook on second side until a dry crust has formed, about 5 minutes longer. Continue cooking, flipping every minute or so, until arepas are mostly dry and relatively stiff, about 5 minutes longer.
Serve hot with cheese, melted butter, or toppings of your choice.