Kansas City, Missouri's COUNTRY CLUB PLAZA was the first regional-class, "car culture" shopping center in the nation. The complex was developed by J.C. Nichols and designed by Edward Buehler Delk, George E. Kessler and Edward W. Tanner. Its first structure, known as the Suydam ["Soo-dam"] Building, opened for business in March 1923. 
Photo from Wikipedia / "Charvex"

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OTHER SIGNIFICANT 1920s SHOPPING CENTERS:

SUBURBAN SQUARE, a fixture in Philadelphia's tony Main Line region, opened in
1928. Originally known as HESTOBEEN SQUARE, the complex was renamed in 1936.
Strawbridge & Clothier opened one of the nation's first branch department stores
here in May 1930.
Photo from the Journal of the Society of Architectural Historians / "The Diffusion of
the Community Shopping Center Concept during the Interwar Decades" / Richard Longstreth

WESTWOOD VILLAGE an on-street collection of shopping, dining and entertainment venues, opened -in the western environs of Los Angeles- in 1929. The domed building on the right originally housed offices for the Janss Investment Company, who developed the shopping center.
Photo from Hoover (Thelner B.) and Hoover (Louise E. B.) photographic collection / https://digital.library.ucla.edu  

Greater Cleveland's SHAKER SQUARE was a de facto downtown for the Shaker
Heights planned community. The shopping center was serviced by the Shaker
Rapid, a streetcar network connecting the southeastern suburbs with downtown
Cleveland. The line began revenue service in April 1920, with the first SHAKER SQUARE stores opening in October 1929.
Photo from https://www.shakersquare.net