If you're not the supervisor or in management, you have no business making business decisions, so choose a situation for which you have the authority to deal with the issue. For example, if you sell faucets and the only one you have left is the display, you could call another store that has some and get one sent to you. You wouldn't delegate that problem to a "proper channel" because YOU are the proper channel.
If it is an issue that you do not have the authority to solve, what you want to do is get the name of the person who has the problem and his/her phone number. Call your supervisor and if he/she isn't there, leave a message about the situation and give the person's name and number. The customer will have to be patient because there's nothing you can do.
Also, tell your co-workers about the situation, and let the next shift be aware of the situation as well. It's all about communication and delegation of authority in business. If you were hired to solve problems, then of course, tell them of your problem solving skills.
Basically, the interviewer wants to get an idea of your judgment and decision making skills, as well as what factors you considered in your decision and whether or not you involved or consulted others. More and more, companies are looking for critical thinkers who use good judgment both for the customer and the company.