With a limited budget, cardboard is the perfect place to start. Trailblazers have created a path, and all you have to do is follow it. Start with Cain's Arcade. Pose a challenge, and watch your learners rise to the occasion. Right away you'll see issues that need solving. How can I ensure kid's are cutting cardboard safely? Use these (cheap) cardboard saws from Canary. How can we easily connect pieces for projects? Make.do sells tool kits for cardboard creations. What else? The sky is the limit.
Collecting is easy, but organizing your collection will take some thought. Consider purchasing containers and designating shelves, sorting items based on your space. I like a basket to hold cardboard tubes from paper towels and toilet paper, and an open shelf to hold boxes of various sizes. Be sure to consider decorating supplies so kids can make it their own.
There is great purpose in play, so expect and encourage children to experiment extensively. Their initial creations will be silly; it will take time to develop a maker mindset with prototypes and models. Be patient and thoughtful.
A final story... my students spent weeks building a cardboard castle. It was not appealing to the eye, but they loved it. I loved their creativity, but I was struggling to ensure their experience was purposeful. These were seventh graders, and it just didn't feel like seventh grade work. It took another student, Molly, to see what I had not. The original creators of the castle had carefully crafted stairs out of cardboard. Molly observed that there castle had very little security features, which was unlikely for a structure of this size and based upon their castle's architecture features. The students problem solved together, and decided they needed a security guard. Insert -- Ozobot. The students carefully diagrammed a path for Ozo to follow, but stopped when they got to the stairs to discuss possible solutions. Swiftly, the cardboard dream castle turned into an engineering problem. They pulled out protractors and rulers, noting immediately that Ozo would need a ramp of the appropriate angle to reach the roof and see out over the kingdom. Cardboard creation became coding, structural analysis, and design thinking. Molly allowed me to see things through a new lens.
Final takeaway: We as teachers gather materials and facilitate experiences, but the kids make the magic. Build them up, talk them through it, and watch and learn!