Combine the pumpkin and vanilla extract, and set aside.
Dissolve yeast and sugar in milk in a large bowl; let stand 5 minutes. Lightly spoon flour into dry measuring cups; level with a knife. Add 2 cups flour, pumpkin mixture, margarine, salt, vinegar, nutmeg, and egg to yeast mixture; beat at medium speed of a mixer until smooth. Stir in 2 cups flour to form a sticky dough. Turn dough out onto a lightly floured surface. Knead dough until smooth and elastic (about 10 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands.
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover dough, and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Coat 24 muffin cups with cooking spray. Punch dough down, and divide in half. Working with 1 portion at a time (cover remaining dough to keep from drying), divide each portion into 12 equal pieces; shape each piece into a ball. Place 1 ball in each muffin cup. Cover and let rise 30 minutes or until doubled in bulk.
Preheat oven to 350°.
Uncover dough; bake at 350° for 20 minutes. Remove from pans; cool on wire racks.