Nitrogen from natural deposits is peachy-keen, but as Dathi Thorfinnson points out, those deposits are quite limited. And, on top of that, guano harvesting is devastating to the usually fairly fragile ecologies of the small, isolated islands that tend to accumulate it.
There are some other minerals that work well for manufacture of ammonia - in particular, sal ammoniac - but, again, we run into the problem of deposits that have limited size and often inconvenient location - for example, one of the larger reserves of sal ammoniac is located in Tajikistan. Like many minerals, sal ammoniac isn't rare per se, as it can be found near most volcanoes, but deposits that are large enough and pure enough to be economical are.
The Haber Process is super convenient. The ingredients - nitrogen and hydrogen - are almost universally available (most factories use natural-gas derived hydrogen, but with even more electricity input, water could do the trick through electrolysis). Where electricity is cheap - which is, more or less, everywhere, depending on the definition of "cheap" - Haber ammonium is awfully handy. Plus, you can build the factory wherever you like, removing the need to ship tons and tons of rock or guano to a processing plant, or build the processing plant in some god-forsaken place and then still ship the tons of ammonia to your transport network somehow.
As far as I'm aware, there aren't really any "alternative processes" for fixing atmospheric nitrogen on an industrial scale. I mean, there are some weird little reactions that predate Haber, but the Haber process is efficient, cheap, and clean, and requires and creates nothing poisonous that needs to be handled with care or disposed of properly.
One idea would be to try and make bioreactors using nitrogen fixing bacteria, like those found in the roots of certain plants, but I suspect that this is somehow unfeasible or it would have been done by now. But who knows? Especially with modern genetic technologies...
Another thing that could help is attempting to recover ammonia from wastewater. Probably the biggest use of ammonia today is in fertilizer, and plants aren't so efficient at taking it up, leading to vast amounts of ammonia in runoff from agriculture. This is terrible for the environment, leading to algal blooms and whatnot, but also represents a major source of waste. Unfortunately, I can't think of any good way to recover that aside from distillation, since ammonia has physical properties quite similar to water and ammonium is a very soluble ion. And while I don't know if distilling ammonia from water is more expensive than production by Haber, it would still take a lot of energy. Plus, we're talking neutral ammonia (very toxic) and not ammonium salts (much less so), so the plant would be pretty dangerous and probably subject to NIMBYism by nearby farmers.