- Do boots really last (only) 400 miles?
In short, yes. If you are a hard-man/woman, you might stretch one pair of boots to half the AT. Normal people go through quite a few pairs - I used 10-ish pairs of trail runners on the PCT, partially because my feet grew 2 sizes and I didn't realize that was why I was suddenly getting blisters from my previously comfortable shoes. So if you know what's up, I think 6-8 pairs for the AT would be reasonable. Boots last a bit longer (as Kevin said, 400 miles maybe), so you might only need 4-5 pairs.
The trouble with boots in that situation is that you generally need to break them in, and a large number of thru-hikers end up with bigger feet within a few hundred miles of starting. So, you can't really buy ahead until you know whether you are the "feet grow" or "feet don't grow" type of person. Thus, using runners is pretty normal - they wear out faster but don't need breaking in.
- Logistically, how do you get more boots as you hike?
On the AT, because of it's popularity, you should be able to identify some gear/shoe stores around the 200-400 miles-in corridor where you could go to try on more boots in a variety of sizes. Alternatively, and what I've done many times, is to only buy one pair before you start hiking.
If your feet stay the same size for the first month, you are probably safe, so have someone at home buy enough pairs for the rest of the trail, and mail them to you via General Delivery at post offices along the trail.
If your feet change sizes, and continue to change, order a range of sizes-widths via the internet (or someone at home) and again, have them delivered via a Post Office. Return all pairs that don't fit. Big sites like Zappos.com will do this no problem (including delivering to post offices and free returns). So once you establish your new size, you walk another 300 miles in your new boots, then order the next pair to the next PO you'll reach. Repeat until done :)