I have one under-the-bed style red cloth organizer box where I store all Christmas-related wrapping items. There are several sections...
On the bottom of the big section is where I store flat-packed boxes (shirt boxes, etc). On top of those is the rolls of wrapping paper we've been using up for 3-4 YEARS. I have not purchased any wrapping paper in AT LEAST three years and we still have several rolls. Then, I have a gift bag with larger gift bags slid into it on top of one half of the large section and I have a shallow box/tray on the other half with tags, pens, scissors, tape, etc in it.
On the bottom next to the big section is a small section and I keep bows and ribbon and fabric bags in there.
On the lid are two clear pouches. I keep the smaller gift bags in one. The tissue paper, small cloth bags, small boxes (DVD holder, gift card holders, etc) are in the other clear pouch.
This whole box holds everything Christmas related in the wrapping department. It also holds Christmas clings and other decor we use upstairs. This box is the only item under our master bed. It is easy to pull out and use. We all take turns using the master bedroom to wrap before Christmas. All other wrapping occasions:
We have an Ikea Ivar shelving system in our family office closet. There is an 8-10 inch gap between the wall and the shelving system. This is where all of our rolls (any kind) go. I have butcher paper here for child drawings, wrapping, packing, etc. I have various craft rolls here, too. I have a plastic organizer hanging from the closet rod that holds the wrapping paper rolls just to keep them off the ground. (It was given to us or I wouldn't have bothered.)
On the Ivar system, I have Trofast bins in some places. One medium bin holds all cards, except birthday and Christmas. A second medium bin holds all birthday items, which includes cards, invites, flat-wrap paper, curling ribbon, gift tags, balloons, confetti, etc. Christmas cards are in a small box next to these two bins. I have not purchased any Christmas cards in 3-4 years and we are finally running out. Once they are gone, we are going green and not sending any paper cards. Once the official "birthday" cards run out, we'll be using the all occasion cards and stamping "Happy Birthday" inside. (I worked at Hallmark back when employees got the leftovers when the store changed the displays. I had a LOT of cards and ornaments!!! I've given away the ones we don't like already.)
I keep a gift bag with all other gift bags inside it on a shelf in the family office closet. I have a plastic shoe box under that bag of bags with all the tissue folded and compressed neatly. I haven't purchased a single gift bag in my entire life. I don't even like them all that much. I do love tissue paper, though, and not just for stuffing gift bags or wrapping (which I like better than wrapping paper!) but also for craft purposes.
Our scissors and tape and pens are already stored in the family office, so it is easy to wrap a gift in here.