Does ammo depletion increase those ammo drops?

There are several different types of ammunition (e.g. sniper rifle, launcher, grenades, etc.). I think being very low on a type of ammunition increases the chance that ammunition of that type spawns in the world, such as from piles and ammo chests. Is this true? If so, does the chance increase at a flat rate relative to ammo amount, or is it that, below a certain amount, ammo of that type is more likely to spawn (at a new static chance)?

When you get low on a certain ammo type, the game does notice that and acts accordingly. From an Inside the Box blog post:

These are the “need” and “emergency” item pools for ammo. These pools, if they are chosen, have a bit more logic that occurs before the items are dropped. Specifically, the game checks to see if you need the item chosen before dropping it. The rules for allowing the drop are based on how much ammo you have. If you are low you will probably get the “need” but not “emergency” ammo. If you are super low you will probably get two extra ammo drops, one from each of these pools. Health works in a similar way that you are more likely to find health when you are low. There are actually some specific lootables that almost always have health when you are low. Have you figured out which ones?

It's a great article to read anyway. Very in-depth look at how loot is generated. A related tip: if you are really low in something like Sniper ammo and come upon a series of ammo boxes: open all of them before taking any. That way, your low ammo state will be in effect for all the boxes and you'll get more.

In my experience, ammunition dropped is random. I use a sniper rife for my main weapon, but I usually found ammunition for weapons I do not need.

That said, my sniper rifle never runs out of ammunition. I find ammunition for it when I have about half my capacity left.