TUCSON, Ariz.—This city, the site of the 2011 mass shooting that nearly killed former Rep. Gabrielle Giffords and sparked a national debate on gun control, is proving how tough it can be for localities to enact new firearms restrictions.

The Tucson City Council advanced a measure on Tuesday that would require background checks on the sale of all guns on city property, an attempt to regulate firearms sales at gun shows held at the Tucson Convention Center. But supporters predict gun-rights advocates will sue to invalidate the...