This is correct. On rare occasions, when the resort is not terribly busy, they preassign rooms and items get delivered prior to the guest check in. This is RARE! For the most part, the runners deliver the items to the room as soon as possible after the guest checks in. This is true for the Disney Florist and all other outside companies.
OP, as someone mentioned, it is possible to do a custom basket through the Disney Florist, but you will still pay a $10 delivery fee on top of the basket cost. The basket cost begins with a custom basket fee plus the cost of the merchandise items you select. At least this is how it worked when I worked for the Disney Florist. I don't think that has changed.
Unfortunately, no one is going to do this cheap. I owned and operated a gift basket company for five years. To do a single small basket involves hours of work. It takes a trip to purchase the merchandise to be delivered. It takes time to put the merchandise into presentation form. Then, the merchandise must be delivered. For me, each delivery took about an hour to do. The cost of gas alone to make the two trips makes it impossible to do this at low cost. I charged $25 plus the cost of the merchandise to be included. After I subtracted the cost of the gas to do the purchase and delivery and the cost of the materials for the presentation, I was lucky to make $5 per basket. That did not account for the cost of my time. In the end, I decided it just wasn't worth it.
I know this is more than what you asked, but I thought it might help to understand why you can't find something delivered at a low cost.