Rinse the kumquats under cool water, quarter them lengthwise, remove the seeds with a knife or your fingers, then chop them to your desired size for jam.
Combine chopped kumquats, vanilla bean, cinnamon stick, sugar, citrus zest, and water in a Dutch oven, cover, and refrigerate for 24 hours to macerate.
Bring the mixture to a simmer over medium-high heat, then reduce to medium and cook, stirring, until thickened or until it reaches 215°F/101°C on a candy thermometer.
Freeze a spoonful of jam on a plate for 5 minutes, then run your finger through it and if it holds its shape and doesn’t flow back together, it’s ready.
Spoon the hot kumquat jam into clean mason jars, let it cool to room temperature, then seal with a lid and refrigerate.