Abbott said the gunman, who lived locally, was likely killed by police officers at the scene, but that was still being investigated.
It's the deadliest school shooting event in Texas in four years after a gunman killed 10 people at Santa Fe High School in Houston in 2018.
As the news broke on Tuesday, thousands voiced their horror and disgust about what happened, while calling for reform of the US's gun laws.
"I'm so sick to my stomach right now! Shooting in an elementary school in Texas. As a survivor of gun violence, teacher, and a mother, we don’t need thoughts and prayers we need policy and change!" one person wrote.
"Texas had a school shooting today but they’re too focused on our uteruses to give a f*ck," another added, referencing the likely overturning of Roe v Wade.
Senator Chris Murphy also pleaded with the Senate in the wake of the event: "14 kids dead in an elementary school in Texas right now," he said.
"What are we doing? What are we doing? We have another Sandy Hook on our hands. What are we doing? What are we doing? What are we doing?"
A number of wounded students were taken to Uvalde Memorial Hospital by ambulance or bus, the facility said in a statement. Another hospital reported that a 66-year-old woman was in critical condition.
Robb Elementary School, located in the small city of Uvalde, has just under 600 students. Some 15,000 people live in the city itself.
It is expected that US President Joe Biden will address the horrific event in the coming hours from the White House.
