A dumpster on private property, such as your driveway or a job site lot, generally doesn't need a Village permit in Hampshire. Things change once any part of the container sits on public ground, like the street, the parkway, a sidewalk, or an alley.
For that, the Village of Hampshire uses its Right-of-Way / Utility Permit Application. The form has a checkbox marked Dumpster, so a container in the street goes through this permit. Along with the application, the Village asks for a refundable performance guarantee, a certificate of insurance that names the Village as an additional insured, a schedule for the work, and a traffic control plan. Because of that paperwork, a contractor often files it.
The permit is handled by Public Works & Engineering, and the Village notes on its permits page that these permits run separately from building permits. The Village doesn't list a fee or a review time for it, so call Village Hall at 847-683-2181 before you pick a delivery date. Village Hall is at 234 S. State Street.
If you aren't sure whether your drop spot is on public ground, your account manager can help you find the right Village contact.