A lot of this depends (no pun intended) on your exact situation as it's going to differ a lot for different people.
My parents are very respectful of my privacy, they haven't gone in my room without significant prior notice since sometime in my teens and when they do it's something for service work with the house and not a full sweep of my room or anything, so I can always shove stuff into a random drawer out of the way until everyone is done. That helps a lot with storing diapers and such without fear of them being found.
To get diapers, I started off with a somewhat elaborate plan though found ways to simplify my methods over time. It was getting close to Christmas, so I started buying Christmas presents off Amazon for my immediate family that was going to be in town. Along with that I bought a pack of Goodnites also off Amazon. A bunch of Amazon packages arrived, all with the excuse of being Christmas presents to justify not commenting further on the contents.
Since then, after ordering a few more normal things that I could actually explain, I came to realize they really didn't ask about what I was ordering, aside from initial curiosity when I first started ordering stuff online, so I was able to drop the whole present framing and just order stuff as long as the packaging didn't give it away. If your family is more nosy you might need something closer to my original plan though, you'd have you gauge your situation for yourself on that.
As for the packaging, it's something you do have to be aware of. Different companies do stuff different, actual ABDL companies tend to be discreet and use a non-obvious name on the shipping label so it looks generic, though you'd want to check their FAQs on a specific company. For Amazon stuff, they have this annoying tendency of shipping stuff in original packaging. They have a button to use an Amazon box, technically, though I had one time where they missed it or ignored it and I was luckily the only person around and was able to sneak the entirely obvious diaper box over to my room. Since then I did a couple more orders with that option checked and was in a normal Amazon box as I wanted, so they may have fixed it, or I just got unlucky the first time. Ultimately, up to you how much you want to take that risk, diapers that don't ship in original packaging are possible to find, but rather rare, especially for finding the right kinds. Sometimes they just have the "ships in original packaging and cannot be hidden" label which is exactly what it sounds like. I just avoid those.
You could also just drive to the stores and get stuff, or have stuff shipped to Amazon lockers or another store, though I haven't done that for a while due to the pandemic. That does have the risk of running into someone you know, so I always like to have an explanation ready, usually something about being for a neighbor or family friend, could also use a nonexistent family member if it's someone you don't know as well. Most people are going to take any explanation as more likely than "Yeah, I felt like buying some diapers for myself because I like diapers" so I wouldn't worry too much about it, just good to have some kind of believable response.
For disposal, I used to just sneak out with them at night after I was the only one awake and put them in the garbage can, typically opening up a garbage bag that was in there, sticking them in, and then closing it, though if the house garbage can was getting full enough I'd just bring that bag out with added diapers. If these weren't going to be an option I just...wouldn't use diapers that day. It's hard to store them for long periods and if I couldn't guarantee I could sneak them out that night I didn't like to take the risk.
I later got a diaper genie which proved to be a great investment. it is bigger and much harder to hide so again, if people are nosy and going in your room you might want to stick to something closer to the previous method. Even with that, I still pretty much use the previous approach but it gives me a way to hold onto them until I get an opportunity and then dispose of the bag at once. (Note: If you're pooping in them they might not be able to conceal the smell fully and you may still need to have a quick way of getting rid of them, I don't do that much so I can't help you there I'm afraid.)
This ended up going a bit longer than I originally planned but hopefully it's helpful. Remember that ultimately your situation might be different, my methods worked for me for almost five years now but I also recognize I'm probably in pretty good situation for this compared to some people.
This also probably isn't the right forum for this, I'd suggest Diaper Talk, but I'm sure a mod can move it if it hasn't already been moved while I was typing this out.
