You need to look into getting some type of range bag. As noted, some use a backpack. Others buy a designed range bag.
Since you mentioned you are looking mostly at ammo, I'll throw this out. If you can find a Harbor Freight store nearby, they have some "tool bags" that are different sizes, and work quite well for ammo. Get their sales flyers & watch for them to go on sale & they get cheap quickly. I use different sizes for different things. I have one just for instructional materials for our WOT clinics, (small stuff,) others for putting smaller boxes of ammo in, (100 round count or 50 round count boxes of centerfire ammo,) or a single bag with nothing but 22 LR ammo, etc. At around $4-$5 a bag, it's easy to get several for specific uses.
I have a seperate range bag that carries all my other stuff besides my ammo.
Now, I also have the Dillon "Border shift" bags. I reload, so I have a bulk quantity of ammo. The border shift is an excellent range ammo bag too.