It is possible but typically not needed except for possibly parts of the Great Divide. If you do need certain supplies, like medicine, refilled, a lot of cyclists mail it to themselves via General Delivery.
Some long-term touring cyclists (touring for 6+ months) very well may ship appropriate seasonal clothing ahead instead of carrying it with them the entire time, but stuff like tubes, tires, etc. can typically be found locally unless you have some unusual situation. This is assuming you do carry at least some extras of the essentials with you, i.e. tubes, patch kits, food, etc.
Remember, most places you can pick up a box has something around, or at least a way to get somewhere to get what you need.
In 40+ years, I have never needed a bump box other than for my meds, which are usually not available in smaller towns and are much cheaper to mail order. However, when I do go on a remote and/or multi-month tour, I do have a few different sizes of the shipping boxes (USPS, Fed-Ex, etc.) at home along with a stock pile of possible items (tires, Gates belt, extra meds, etc.) that the better half can expedite to me if needed. Obviously, if I am outside the USA, then the boxes are international boxes. Again, never had to use it.
Hope this helps. Tailwinds, John