Short Guide
Once Kay and Yu start their seed-collecting contest, make sure you frequently swap between the two, as there is a limit on how many seeds each character can get. Be sure to keep an eye on your inventory to see what seeds are still available for collection.
Long Guide
Kay is the first to find a seed, which starts a contest between the two. There are checkpoints throughout the contest, so you can't get all 18 seeds only on Kay or only on Yu. Kay will collect some, and then you'll never see a seed till Yu grabs one.
Note 1: The first seed Kay picks up will not count towards the contest. So if their score is Kay 5 - Yu 1, then in your inventory, you won't have 6 seeds but 7.
Note 2: The [Total ?] shows what the number of seeds should be in your inventory (excluding the 2 special ones), so if you find this guide halfway through a playthrough, you can find your place.
Play as:
Kay for 5 seeds = K5 - Y0 [Total 6]
Yu for 1 seed = K5 - Y1 [Total 7]
Kay for 1 seed = K6 - Y1 [Total 8]
Yu for 2 seeds = K6 - Y3 [Total 10]
Kay for 1 seed = K7 - Y3 [Total 11]
Yu for 4 seeds = K7 - Y7 [Total 15]
Kay for 1 seed = K8 - Y7 [Total 16]
Yu for 1 seed = K8 - Y8 [Total 17]
Yu or Kay for the final Victory seed [Total 18]