Though it's obvious, and probably not your first choice, you might resort to buying them if you need to find moving materials quickly. Office supply stores, shipping companies and specialty stores online sell moving boxes, packing paper and moving kits that also include items like rolls of packing tape and bubble wrap.
Even if you don't have enough old boxes sitting around your garage or have a friend who has recently moved, there are a few ways you can get cardboard boxes for free. If you live in a suburban area with shopping centers and stand-alone stores, you can collect boxes by driving behind the stores to find ones they are getting rid of. Some of them may be a little damaged or a few could have something spilled on them that make them unusable. But stores are constantly unpacking new stock from boxes and throwing out boxes that are in perfect shape to be used again. Office complexes are another place to look.
Other people are also moving their homes and offices and might be pleased to have you take the boxes they have used off their hands once they are finished with them. There are online communities where people post things they are giving away for free, including moving boxing, like Freecycle.org and Craigslist.org, where people may also be selling moving materials at lower rates than stores.
Old newspaper is often readily available at no cost. It makes a suitable paper to wrap many items, although it might not be quite as durable as specially made packing paper, and it might, particularly on hot days, rub some ink off on the items that are wrapped in it. So it may not be the best material to use for wrapping extremely delicate or valuable items.
There are also environment-friendly alternatives, like using Rent-A-Green Box, a service that rents out boxes made from plastic waste mined from landfills, delivering them a week before the move in green trucks powered by waste vegetable oil and bio-fuel. The trucks return to pick them up two weeks later. The service also sells other environment-friendly moving materials. Though services are spreading, they are not available everywhere.