Back in Lincoln's day, all they had to kill vampires were axes, unlike today's kids who get tasers and light sabers. Abe's ax starts out pretty humble. The first time he wields it, it's to chop wood, which is a punishment for not hunting (he had a bad run-in with a messy turkey). At first, Abe can barely lift the thing. But the exercise soon makes him strong and eventually he's better at chopping wood than even his dad (1.46-8).
There are no vampires yet—at least, as far as Abe knows—but the ax already has significance. It's a symbol of how strong and skilled Abe is. That connection only grows deeper as Abe learns about vampires and starts training. By the time he's done, "He could bury the head of an ax in a tree from over thirty yards" (3.3).
That last part may not be based on fact, but there is a story about President Lincoln chopping some wood with an ax to prove how strong he was. Lesson learned: ax swinging takes a lot of strength.
Once Abe gets introduced to vampires (Abe, meet vampires, vampires, meet Abe), the ax becomes his primary weapon against them. He always carries the same weapons with him: ax, crossbow, and his special-made matches called his "martyr" (4.18). But the ax is really his most-used weapon. In fact, the ax gets mentioned in every chapter—though in Chapter Twelve it's only mentioned in a caption for a picture. By contrast, his crossbow is only mentioned in five chapters. Poor crossbow.
When Abe is hunting vampires, the ax is kept sharp: "He sharpened his ax until the blade 'blinded all who looked upon it'" (2.95). When Abe retires, "The blade of his ax was allowed to rust" (9.16). When Abe comes out of retirement, he "scraped the rust off his ax" (10.65).
That means we can chart how much he's fighting vampires by the state of his ax. And when Abe becomes a vampire, he dreams of hunting vampires with his ax. Well, of course he does—if he's hunting vampires, it's going to be with his ax.
But what's interesting about this mention is how friendly Abe is with his ax: "Over the years, the handle had been worn into the perfect companion for his massive hands. Each furrow a welcoming friend. It was hard to know where he ended and the ax began" (14.7). Abe has had several vampire hunting friends, from Edgar Allan Poe to Ward Hill Lamon. But the friend who stays with him the most is his ax.
But why does Abe use an ax to kill vampires? Why not something more military-esque like a sword? Maybe that's the answer: a sword only is for violence, but an ax can be used to cut off vampire heads or to chop firewood or to cut rails to build a fence or cabin.
So maybe Abe uses an ax because axes can be used to destroy or to create. When Abe is hired by James Gentry to sail down to New Orleans, Abe "put his ax (and in fairness, the carpentry skills he'd learned from his father) to work building a sturdy, forty-foot flatboat from green oak, cutting each plank and fastening it to his frame with wooden pegs" (4.25). So that ax comes in awfully handy. It's a multipurpose tool in the way that Abe is a multipurpose man—businessman, flatboat sailor, lawyer, politician, president… vampire hunter.
Abraham Lincoln the man (as opposed to the character) really was associated with an ax. It's all over the pro-Lincoln posters and drawings, which often show him wielding one. Check out this cartoon, in which Abe holds an ax with the words "Emancipation Proclamation" written on it. And here's Lincoln moving in to the White House, carrying a bag and an ax (just the essentials, of course). If only all those cartoonists knew what that ax was really for.