Found four articles on a "Vargas Baseball Club" near San Antonio 1949-51.

The first, San Antonio Light 2-12-50, indicates the Vargas team has reorganized, " The club still under the sponsorship of Vargas Place, located on the New Laredo Hwy, but will be managed by Charles A. Gallegos, a pitcher of the past season of the Vargas club. The club is inviting all within a radius of 300 miles to schedule games."

The second, San Antonio Express 5-1-50, announced a pair of wins for Vargas over the Balboa Eagles at Vargas Field, 12-2 & 4-1. Winning Vargas pitchers were Benny Ortega & Manuel Daniel.

The other two, were from 1951 but just mentioned contact info (Charles A. Gallegos) and some Amateur baseball notes on the "Spanish American League" San Antonio Light 3-25-51.

Unlikely this is the team from the pinback... but may be helpful.