In a spice grinder, coffee grinder, or Vitamin Dry Blade grind the rice, almonds, and cinnamon in batches until the mixture resembles a fine powder. Place in a 2-quart container with a lid. Add water, place the lid on the container, and shake to combine. Allow to sit at room temperature for 12 hours.
Add sugar, lime juice, and vanilla to the mixture. Blend in batches on high until the mixture is nice and smooth. Pour the mixture through a fine-mesh strainer lined with a flour sack towel or cheese cloth placed over a bowl. Use a rubber spatula to press the solids to extract all of the liquid. Discard the solids, pour into a container until ready to serve.
When ready to serve, fill a punch bowl with ice and add rum and horchata then stir to combine. Ladle into ice filled glasses and drop in a small piece of cinnamon stick in each glass and top with a slice of lime.