Go to the website of your county government. County auditors and tax assessors are most likely to track the map number of your property, so check to see if there is a section of the website devoted to these departments. A section titled "Property Search" or something similar might be an option too. For example, in Greenville county there is a tab that says "Popular Services" and a drop down box that says "Real Estate" which leads to the option "Property Search."
Type in your address. An exact address will return more accurate search results than just a street name. Counties can also give you the option to search for a property by owner name.
Choose your address from the search results. Many counties return search results presenting the property address, owner, and map number, so the number may be available at this step. The term "parcel ID" or "parcel number" can be taken to mean map number in most circumstances.
Find the map number in the property description. Different counties provide different levels of information about a property. In many cases the map number will be displayed prominently on the first page, but in some cases you may have to choose other tabs to find the number.