For offices where no candidate was nominated by an established political party that party can appoint someone to the ballot for the General Election so long as they submit petitions and their candidate filing papers by May 31st. For example in the 1st State House District the Democrats nominated incumbent Dan Burke, he won his primary with no opposition but no Republican candidate filed so the party could appoint someone to appear on the fall ballot so long as they submit the required paperwork. That is true for every such situation.
Here are the relevant sections of the election code which explain this process. I had to look it up so I figured others might find it useful as well.
10 ILCS 5/7-61 If the name of no established political party candidate was printed on the consolidated primary ballot for a particular office and if no person was nominated as a write-in candidate for such office, a vacancy in nomination shall be created which may be filled in accordance with the requirements of this Section. If the name of no established political party candidate was printed on the general primary ballot for a particular office and if no person was nominated as a write-in candidate for such office, a vacancy in nomination shall be filled only by a person designated by the appropriate committee of the political party and only if that designated person files nominating petitions with the number of signatures required for an established party candidate for that office within 75 days after the day of the general primary. The circulation period for those petitions begins on the day the appropriate committee designates that person. The person shall file his or her nominating petitions, statements of candidacy, notice of appointment by the appropriate committee, and receipt of filing his or her statement of economic interests together. These documents shall be filed at the same location as provided in Section 7-12. The electoral boards having jurisdiction under Section 10-9 to hear and pass upon objections to nominating petitions also shall hear and pass upon objections to nomination petitions filed by candidates under this paragraph. ... In the proceedings to nominate a candidate to fill a vacancy or to fill a vacancy in the nomination, each precinct, township, ward, county or congressional district, as the case may be, shall through its representative on such central or managing committee, be entitled to one vote for each ballot voted in such precinct, township, ward, county or congressional district, as the case may be, by the primary electors of its party at the primary election immediately preceding the meeting at which such vacancy is to be filled.10 ILCS 5/8-17 In the event that a candidate of a party who has been nominated under the provisions of this Article shall die before election (whether death occurs prior to, or on, or after, the date of the primary) or decline the nomination or should the nomination for any other reason become vacant, the legislative or representative committee of such party for such district shall nominate a candidate of such party to fill such vacancy. However, if there was no candidate for the nomination of the party in the primary, no candidate of that party for that office may be listed on the ballot at the general election, unless the legislative or representative committee of the party nominates a candidate to fill the vacancy in nomination within 75 days after the date of the general primary election. Vacancies in nomination occurring under this Article shall be filled by the appropriate legislative or representative committee in accordance with the provisions of Section 7-61 of this Code. In proceedings to fill the vacancy in nomination, the voting strength of the members of the legislative or representative committee shall be as provided in Section 8-6.