Whisk coconut milk in a large, heavy saucepan until smooth. Sprinkle agar-agar flakes on top. Let sit, undisturbed, until flakes are rehydrated, about 5 minutes.
Heat the milk gently over medium-low heat, stirring constantly, until a thermometer inserted into the center reads 180 degrees F (82 degrees C). Reduce heat to low and cook, stirring occasionally, until agar-agar is fully dissolved, about 5 minutes.
Remove the saucepan from the stovetop. Let milk cool, stirring occasionally, until it reaches 115 degrees F (46 degrees C). Speed up the cooling process by placing the saucepan in an ice water bath; stir continually until it reaches the correct temperature.
Open the probiotic capsules and sprinkle the powder over the milk. Discard capsules. Stir in sugar until dissolved.
Transfer the yogurt mixture to sterilized glass jars and cover with lids, or leave in the saucepan and cover. Let sit in a warm place, about 110 degrees F (43 degrees C), until thickened to desired consistency, 12 to 24 hours.
Chill yogurt in the refrigerator until set, 4 to 6 hours.