Their borders are controlled but only on one side by Israel. Why aren't they using the other border for things like mail. Or for everything.
My understanding is that Israel controls borders and the flow of goods and people in both Gaza and the West Bank and block whatever they please. Do Palestinians have exclusive control over any external or internal border? I'm not aware of this and doubt it since this would be dangerously close to granting Palestine an independent state, and Israel is not about to do that.
For what it's worth, the Wikipedia article seems to describe Israel as in control and capable of overruling any Palestinian decision at the crossings. https://en.wikipedia.org/wiki/West_Bank#Crossing_points
I could see it for the crossings into Israel but how would they control the crossings in to Egypt and Jordan.
You're right; my post was incorrectly about Gaza and not the West Bank. Gaza definitely is a more perilous situation given it is so small and cutoff from the outside world. That is not to say the West Bank is markedly different, but merely that there are at least many cities and a larger territory, so there can be at least some commerce and opportunities.
In both cases all borders--whether with Israel or neighboring countries--are controlled. Shortages of fuel and medical supplies occur. Roadblock checkpoints exist between cities to slow travel and limit freedom of movement. A significant portion of the population are refugees whose homes have been destroyed or who have been forced out of their homeland. The correct and neutral word to describe the situation in the West Bank truly is military occupation.