Starbucks uses a machine called the Mastrena. It is made in Switzerland exclusively for Starbucks and is not sold publicly. You can buy Starbucks brand espresso beans to get closer to the in-store coffee experience.
Before the Mastrena, Starbucks used La Marzocco Linea. These were standard espresso machines with a removable portafilter, which required the tamping process.
After Starbucks' explosion in popularity during the 1990s, the company transitioned into the Mastrena machines. The Mastrena is a superautomatic machine, meaning the traditional portafilter and tamping process is not necessary. Superautomatic machines speed up coffee making and allow anyone to operate the machine with very little training, but they produce a less flavorful cup of coffee.