Is it possible I deliver my products there myself in my own car?

I have only heard of a couple 7 or 8 figure gross sellers doing this, otherwise it's cost prohibitive.

If it is, what is the procedure?

You'd have to start by incorporating a shipping carrier company, get bonded and commercial transport insurance, make your delivery vehicle pass an annual commercial safety inspection with a BC CVSE approved inspection facility, you will also need to get a NSC (national safety code) number for your shipping company's fleet.

You will also need to get PPE (hard hat and safety vest and steel toe boots for Amazon's yard), and gain the ability to book delivery appointments on Amazon's carrier website. They may request an appointment anytime of the day or night, so you might get stuck with dropping off in the middle of the night.

just a few km from my place..

There is no guarantee that Amazon will give you the Fulfillment center closest to you. The one on Braid St in new west only usually deals with small and light items, and returns.

You might be stuck driving it to one of the other ones in BC, Alberta, or on the other side of the country back east. You don't get to pick, Amazon picks the fulfillment center that is best able to handle your inventory and the location where your products are in the most demand.