The Constitution
Federal election laws like HAVA, limit their mandates to apply to federal elections—congressional and presidential. It may seem silly to some that Congress can only pass election laws that apply to federal elections, unlike in other areas of law where federal law preempts state law. I frequently get asked about this so let me lay the constitutional and jurisprudential background out. In short: it's the Constitution, stupid!
The Constitution
The powers to regulate congressional elections between the federal and state legislatures is spelled out in Art. 1, Sec. IV, cl. 1 of the Constitution:
The times, places and manner of holding elections for Senators and Representatives, shall be prescribed in each state by the legislature thereof; but the Congress may at any time by law make or alter such regulations, except as to the places of choosing Senators.
And the States appoint electors to elect the President while the Congress can choose a day for them to select the President (Art. 2, Sec. 1):
Each state shall appoint, in such manner as the Legislature thereof may direct, a number of electors, equal to the whole number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or person holding an office of trust or profit under the United States, shall be appointed an elector. [...] The Congress may determine the time of choosing the electors, and the day on which they shall give their votes; which day shall be the same throughout the United States.
Well, that's about as thick as mud, right?
How have the courts over the years interpreted this language? The Supreme Court has held that Congress has explicit authority to regulate congressional elections and a "broad implicit authority" to regulate presidential elections. In addition to Art. 1, Section IV, clause 1 there are a number of important Supreme Court decisions here; two for congressional election regulation (Smiley v. Holm, 285 U.S. 355 (1932); U.S. Term Limits, Inc. v. Thornton, 514 U.S. 799 (1995)) and one for presidential election regulation (Burroughs and Cannon v. United States, 290 U.S. 534 (1934)).
Congressional power over congressional elections
In Smiley, the Court was presented with the question of what does the text of Art. 1, Sec. IV, cl. 1 actually mean. It found that the U.S. Congress has full supervisory power over congressional elections (Smiley at 366-367):
[T]he second clause of article 1, 4, [provides] that 'the Congress may at any time by law make or alter such regulations,' with the single exception stated. The phrase 'such regulations' plainly refers to regulations of the same general character that the legislature of the State is authorized to prescribe with respect to congressional elections. In exercising this power, the Congress may supplement these state regulations or may substitute its own. It may impose additional penalties for the violation of the state laws or provide independent sanctions. It 'has a general supervisory power over the whole subject.'
In Thornton, the voters of Arkansas passed a constitutional amendment that altered the qualifications for AK representatives to the US. House. That was found to be unconstitutional as the Constitution prohibits states from adopting additional or conflicting qualifications.
Congressional power over presidential elections
As to power over regulating presidential elections, Burroughs recognized that Congress' power to regulate presidential elections was implicit compared with the powers over congressional regulations.
For presidential elections, the Constitution explicitly gives states the power to choose electors (Art. 2, Sec. I) and the manner that they're elected, and gives Congress the power for choosing the time and day of the election (with the day being the same throughout the U.S.). However, the Court said reading Congress' powers as being that narrow is "without warrant" and that there is a certain amount of regulation that Congress must be able to engage in over Presidential elections. They need to be able to protect the process and integrity of that election especially because states are be poorly positioned to do so or would necessarily do so inadequately.
In short, the Constitution implicitly gives this power (Burroughs at 545):
The importance of his election and the vital character of its relationship to and effect upon the welfare and safety of the whole people cannot be too strongly stated. To say that Congress is without power to pass appropriate legislation to safeguard such an election from the improper use of money to influence the result is to deny to the nation in a vital particular the power of self-protection. Congress, undoubtedly, possesses that power, as it possesses every other power essential to preserve the departments and institutions of the general government from impairment or destruction, whether threatened by force or by corruption.
and that how Congress chooses to regulate presidential elections is solely within their judgment (from 547-548):
The power of Congress to protect the election of President and Vice President from corruption being clear, the choice of means to that end presents a question primarily addressed to the judgment of Congress. If it can be seen that the means adopted are really calculated to attain the end, the degree of their necessity, the extent to which they conduce to the end, the closeness of the relationship between the means adopted, and the end to be attained, are matters for congressional determination alone.
It's the Constitution, stupid!