The Daily Herald offers this report (link via Politicalwire.com), which begins:

State law doesn’t cover how to replace Senate candidate dropouts
The buzz among Republican insiders is several party leaders have spoken to U.S. Senate nominee Jack Ryan and asked him to get his divorce file situation straightened out sooner rather than later.
Two Chicago media outlets are asking a California judge to release sealed documents Ryan’s ex-wife’s attorney has alleged contain information embarrassing to Ryan’s political career. Ryan is fighting the documents’ release, saying he wants to protect his son.
Here’s a new wrinkle: If the documents are released and the information proves fatal to Ryan’s campaign, election law makes no specific provision for who appoints a new Republican U.S. Senate candidate.

The article does not mention one of the trickiest issues related to this question. Because this is a race for a federal office, the United States Constitution may limit how a replacement may be named. Article I, Section 4 of the Constitution provides that the state legislature may prescribe the rules for picking Senators. If the legislative scheme has a gap, can courts (as opposed to the Illinois legislature) fill it in without violating Article I, Section 4?
The concurring opinion of Justice Rehnquist in Bush v. Gore (joined by Justices Scalia and Thomas) raised a parallel question under Article II (pertaining to state legislatures providing the rules for choosing presidential electors). But an attack on the New Jersey Supreme Court’s decision to allow late replacement of dropout Robert Torricelli in the New Jersey Senate race a few years ago on these grounds failed, and the United States Supreme Court declined to intervene. The issue is far from settled.