I realize that states get to run their elections. However the congress has passed many laws regulating elections and voter access.
They passed the voting rights act for example, though it was only just recently struck down, but not on the grounds of constitutionality but on more technical grounds.
They've also passed laws that require polling places to be accessible to the disabled.
And there is the national voter registration act which more or less mandated access to voting registration and put regulations regarding state voter rolls maintenance.
Of note the voting rights act wasn't struck down because the SCOTUS thought it was unconstitutional for congress to regulate state elections. They struck it down for a more specific reason.
"On June 25, 2013, the Supreme Court
ruled that the coverage formula in Section 4(b) of the Voting Rights Act — which determines which jurisdictions are covered by Section 5 — is unconstitutional because it is based on an old formula. As a practical matter this means that Section 5 is inoperable until Congress enacts a new coverage formula, which the decision invited Congress to do. " [However congress has yet to move on this]
The Voting Rights Act was passed in 1965 to ensure state and local governments do not pass laws or policies that deny American citizens the equal right to vote based on race. On June 25, 2013, the Supreme Court swept away a key provision of this landmark civil rights law in Shelby County v. Holder.
www.brennancenter.org
Section 4 was apparently a formula derived in the 1960s for identifying places that the voting rights act would regulate and section 5 is the protections it confers (requiring DOJ oversight of election law changes). The SCOTUS kept section 5 and congress has yet to more or less re-write section 4 it appears.
This is what Kennedy said at the time about the decision. "“Our country has changed,” Chief Justice John G. Roberts Jr. wrote for the majority. “While any racial discrimination in voting is too much, Congress must ensure that the legislation it passes to remedy that problem speaks to current conditions.”
So clearly there is room for congressional regulation of voting rights and access to me and its mostly a matter of a willing congress ready to say enough is enough.