Do you want to gift your shares, or just want the niece to own some shares of VTI (or any other security)?
If the former, you need to contact your broker, and you will need to work with your niece (or parent) to start her own account.
If you just want to give her $1000 so she can start her own account, you can do that.
Or you can use Sparkgift.com to send her $1000 in VTI shares. They will even do fractional shares.