Stir pumpkin and sugar together in a saucepan. Cover and let stand, 8 hours to overnight.
Place unpeeled lemon wedges in a food processor; puree until smooth.
Place pumpkin and sugar mixture in the saucepan over high heat. Stir in lemon and ginger. Bring to a rolling boil; cook, stirring frequently, until pumpkin is translucent but still holds its shape, about 30 minutes. Mash to desired consistency.
Bring a large pot of water to a boil. Add jars and lids; boil until sterilized, at least 5 minutes. Pack pumpkin mixture into hot jars, filling to within 1/4 inch of the top. Wipe the rims of the jars with a moist paper towel to remove any residue. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars 2 inches apart into the boiling water using a holder. Pour in more boiling water to cover jars by at least 1 inch. Bring the water to a rolling boil, cover the pot, and process for 10 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool, 12 to 24 hours. Press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.