I noticed there’s a verified run in this category that has pool cleaners. So are you allowed to buy shop items if you don’t use them?

well in other category's like seeing stars you can have up to 10 slots if buy them so i think it only applies to plants and the rake

Actually, for levels you do choose your plants, you must not buy any extra seed slot because this will affect your final time, so for shopless, you must have only 6 seed slots.