Carter said the reports were only adding "unnecessary pain to a grieving family".
"To this point, no one has been authorized to speak on behalf of the family regarding any personal details surrounding Sunday's tragedy, including stories related to the family's previous air travel decisions," Carter said. "We ask members of the media for respect and responsible judgment during this difficult time."
Meanwhile, in her social media post, Vanessa thanked fans for their support.
"My girls and I want to thank the millions of people who've shown support and love during this horrific time," she wrote. "Thank you for all the prayers. We definitely need them.
"Thank you for sharing your joy, your grief and your support with us. We ask that you grant us the respect and privacy we will need to navigate this new reality."
The Lakers - Bryant's team in his entire 20-year NBA career - are set to play their first game since the crash.
The team plan to hold a tribute to Bryant at the game.
Lakers Vice President Rob Pelinka, who served as Bryant's agent during much of his playing career, said the loss of his "best friend" and "sweet goddaughter" was tough.
"He was, simply put, the most inspirational athlete of our time. What the world may not know is that he was also the best friend anyone could ever imagine," Pelinka said in a statement. "Everything Gigi stood for, I am so proud of."
The Lakers play the Portland Trail Blazers at 4:30pm NZT.