My limited understanding:
The green leaf is grown either on centralized plantations with their own tea processing facilities, or by small farmers who then sell their leaf to tea processing factories. The factories process the leaf into made tea, grade it, and pack it up in paper or plywood chests. Most of the tea is then sent to an auction center, where it is bid on by corporate buyers who plan to use it in their commercial blends. The corporate buyers blend the tea together with a bunch of other teas to create their standard tea flavor (P.G. Tips, Lipton's Yellow Label, etc.), which are then sold to grocery chains in bulk for stock in their stores.
However, there are many variations on this.
For instance, sometimes the tea is sold directly from the factory, skipping the auction system. Sometimes a government will be allotted tea as part of a trade agreement with the producing country. (India had such an arrangement with the USSR before its collapse.) Although the auctions are dominated by the corporate blenders, other wholesale buyers also have the opportunity to purchase tea, some of which is then marketed unblended as "single estate" tea by specialty teashops and smaller tea sellers. Sometimes estates will enter into "futures" agreements, where a buyer agrees to purchase the entire crop of an estate before it is grown.
I suppose the short answer is that while there are a few common ways that tea gets to stores, there isn't a single way.