Place lime juice, vinegar, and sugar, in a small saucepan over medium heat. Bring to a boil, stirring occasionally. Remove from heat, add lemongrass and lime leaves; steep until cool. Strain through a fine mesh sieve, discarding solids, and set aside.
Heat oil in a medium skillet over medium heat. Add shallots and ginger, and cook, stirring, until almost tender, about 30 seconds. Add kumquat slices and reserved lime juice mixture. Bring mixture come to a simmer and cook for 30 seconds. Season with salt and let cool slightly before using.