The family lived in Prosper before moving to Lakeway, Davis said.
Authorities did not immediately confirm their identities.
The Copelands were celebrating a relative's birthday when the attack happened, family member Haley Copeland wrote on Facebook.
"This is an extremely difficult time for my family and anyone who knows Sean and Brodie Copeland," she said. "Losing a loved one is hard no matter the circumstances but losing a loved one in such a tragic and unexpected way is unbearable."
"I don't even know how to put this in words," wrote relative Heather Copeland in a post that appeared on Twitter. "Today was a very (tragic) day for my family. ... Please keep my family and all the other (families) affected by the tragic event that occurred tonight in your prayers."
Brodie played youth baseball, the Austin American-Statesman reported, and his father coached youth baseball.
The Hill Country Baseball Academy, located in Dripping Spring, Texas, asked for prayers on behalf of the Copeland family. Brodie played baseball at the academy.
"Nobody deserves this type of fate, especially not such a wonderful family," the academy wrote on Facebook. "You are in our hearts, thoughts and prayers. Rest in peace, Brodie and Sean. You will be remembered by many."
According to French officials, dozens of people died just after the Bastille Day fireworks finished in Nice Thursday night.
Nice Matin journalist Damien Allemand described a chaotic scene as the truck struck multiple people.
"An enormous white truck came along at a crazy speed, turning the wheel to mow down the maximum number of people," he said. "I saw bodies flying like bowling pins along its route. Heard noises, cries that I will never forget."
The investigation into the attack is ongoing.
The Associated Press contributed to this report.