An Arkansas contractor taking work in Illinois deals with the same four questions everywhere: who licenses the trade, when a licence is actually required, when workers compensation kicks in, and where the business registers. Below is Illinois's answer to each, with the agency's own link.
Licensing
IDFPR (roofing); Illinois Department of Public Health (plumbing)
No statewide general-contractor license; GC licensing is local. The state licenses roofing contractors via IDFPR (Roofing Industry Licensing Act, 225 ILCS 335) and plumbers and plumbing contractors via the Illinois Department of Public Health (225 ILCS 320).
Workers compensation
Illinois Workers' Compensation Commission
Near-universal: even one part-time employee triggers the requirement to obtain workers' compensation insurance; no employee-count threshold.
Business registration
Illinois Secretary of State, Department of Business Services
https://www.ilsos.gov/departments/business_services/home.html
New hire reporting
Most states require newly hired employees to be reported within a short window of their start date.
https://ides.illinois.gov/employer-resources/taxes-reporting/new-hires.html
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Starting or running the business itself is covered in the contractor business guide. Arkansas licence records for Benton and Washington counties are in the contractor directory.