Add the pumpkin puree, brown sugar, pumpkin pie spice, and vanilla extract to a medium saucepan over medium-low heat. Whisk everything together until it's smooth and starts to smell fragrant, which usually takes about 2 to 3 minutes. This step is really about waking up the spices, so don't rush past it.
Pour in about 2 cups of the apple cider and drop in the cinnamon sticks, then stir well to combine. Let the mixture come to a gentle simmer for 2 to 3 minutes, but don't let it boil hard or the pumpkin can turn grainy. Once it's warm and well blended, take it off the heat.
Let the pumpkin base cool to room temperature, then run it through a fine mesh strainer if you want a smoother, restaurant-style punch. I usually skip straining for a casual get-together since the texture doesn't bother most guests. Pop it in the fridge for about 30 minutes if you're short on time before serving.
Pour the cooled pumpkin base into your punch bowl or a large pitcher, then add the remaining apple cider and the orange juice. Stir well so the pumpkin puree fully blends into the liquid instead of settling at the bottom. Taste it here and adjust the sweetness with a little extra maple syrup if needed.
Right before serving, pour in the chilled ginger ale or sparkling water so the punch stays as bubbly as possible. Give it one gentle stir, being careful not to knock out all the fizz you just added. If you have time, chill the punch bowl itself beforehand so the drink stays cold longer.
Add a good amount of ice along with the apple slices, orange wheels, and reserved cinnamon sticks on top. I like to float a few extra apple slices right before guests arrive so they still look fresh and bright. Ladle into glasses and serve immediately while it's still cold and fizzy.