I like to carry all that stuff in my scooter too, I use the little basket on front but if you have more stuff you can get a bigger basket that fits on the back of the scooter, most rental places have them available as an add on to your rental.
I use a backpack sometimes for the stuff I want to keep closer, you can hang it over your seat and then take it with you when you leave the scooter. You can use something you already have, a cheap one from a big box store, or even get one made for scooters (do a web search for 'mobility scooter backpacks' to see several).
Have Fun!
a lot of people use regular backpacks and just bring it with them when they leave the scooter.
If you are bringing the scooter with you into lines, you will usually have the scooter until boarding the attraction. I used to worry about DD's backpack on her wheelchair, but don't usually now. In all the times we've left it, we've never seen any evidence that it was disturbed or moved.
We don't have anything of great value in it, but since the wheelchair is parked inside the attraction, people walking by it are mostly looking for how to get out of the attraction and most don't even 'see' the mobility device.
Even if they do, they don't know who it belongs to, so would be taking a chance that the owner might be right behind them.
For shows, you will usually bring the scooter right in and sit on it during the show.
I am led to believe that a plastic grocery bag (not a Disney or Universal or Sea World shopping bag which someone may think contains souvenirs) would be less likely to be disturbed compared with a back pack. Use two (double layer bagging) to reduce the chance of tearing or breakage.
A plastic grocery bag would be a poor choice. For one thing, the handles would not fit well on the frame of the scooter to hold the bag in place.
If the handles are used as 'straps' to hold it on, they would also hold the bag open. That would make it more likely for someone to pick something out of the open bag or even for things to fall out of the bag. Because it's not at all secure, it would be easier for anyone to remove something from it than a backpack or other bag.
Even a doubled plastic shopping bag would be likely to tear or break, leaving nothing to contain your belongings. And, there is nothing about a grocery bag that would make it less likely to be removed from the scooter and stolen. In fact, it might be easier for a thief to take; they could just put it inside of their own bag and thrown it away later after they remove anything of value.( much easier to hide than a stolen backpack).
Plus, one thing people don't think about if their bags are left outside is birds and squirrels. People have posted pictures online of birds and squirrels going into bags on strollers to get at food. At least if you have a backpack or other closed bag, it makes it a bit more difficult for animals to get in.