In Illinois, state's attorneys are elected by each county for a four year term. This is the norm in the US: 47 states and the District of Columbia elect state's or district attorneys. Of the remaining three, only in Alaska does the governor appoint the state's attorneys; in Connecticut they are appointed by a commission, and in New Jersey they are appointed by the court system.
District/state's attorneys (which are largely the same thing) have a great deal of autonomy. The governor cannot generally remove them from office directly, and funding for state's attorneys offices mainly comes from the legislature or federal grants, so governors can't effectively use the power of the purse over them. The governor's office can (of course) contest a d/s attorney's decisions in court, and governors have a lot of in-state influence and power that can make challenging them a risk (in terms of re-election and such). But at the end of the day, the d/s attorney gets to decide what cases s'he will and will not pursue, and short of obvious malfeasance (overt bias, influence peddling, or other activities that could be prosecuted) there isn't much to be done except to elect someone else in the next cycle.
Of course, there is some legal gray area when it comes to things like citations, fines, and other municipal actions. These are usually handled bureaucratically, without the involvement of the state's attorney's office. So while a state's attorney might refuse to prosecute a stay-at-home order, this won't necessarily stop police from issuing fines or judges from enforcing them; it would only stop higher-level prosecutions (whatever those might be).