Galvis was born November 14, 1989, in Punto Fijo, Falcón, Venezuela, and at age 14, the Philadelphia Phillies began to scout him during his days playing youth baseball. He participated on Latin America's team in the Little League World Series, but due to his slight stature (5 feet 10 inches (1.78 m) 154 pounds (70 kg)), struggled to captivate scouts' attention. Galvis said, "They told me that I couldn't play (pro) baseball. But I knew I had the ability." In 2006, scouts Sal Agostinelli and Jesus Mendez signed him during the amateur player signing period. Galvis, 16 years old, was not a particularly heralded prospect, but did receive some attention from scouting experts before signing with the Phillies.
Galvis participated in the Florida Instructional League in 2006. The next season, he officially began affiliated baseball in 2007 with the Williamsport Crosscutters at age 17, as one of just five players not born in the United States. He made an impression with Phillies' personnel due to outstanding defensive skills that drew comparisons to those of Baseball Hall of Famer Omar Vizquel, and also demonstrated solid instincts while baserunning, despite struggling to get on base. Concurrently, he hit .203 with nine stolen bases in 38 games. In 2008, with the Lakewood BlueClaws in the South Atlantic League, Galvis hit .238 with 14 stolen bases in 127 games. He had a particularly strong May, during which he hit .313 with 14 RBIs. He also earned placement on the league's postseason all-star team. He began 2009 with the Clearwater Threshers in Class A-Advanced, and was the Phillies minor league player of the week after he hit .417 from April 20 to 26. The following May, he fractured his right ring finger, and spent more than two months on the disabled list. Thereafter, he spent seven games with the GCL Phillies before finishing the season at Double-A, where he hit just .197.