最后用以下代码解决了这个问题：

List<string> packagesToInstall = new List<string>();
packagesToInstall.Add("SomeNuGetPackageIdentifier");
var componentModel = (IComponentModel)Microsoft.VisualStudio.Shell.ServiceProvider.GlobalProvider.GetService(typeof(SComponentModel));
IVsPackageInstaller2 installer = componentModel.GetService<IVsPackageInstaller2>();

//Loop through the list of Strings and install the Packages.
foreach (string package in packagesToInstall)
{
    try
    {
        installer.InstallLatestPackage(null, vsProj.Project, package, false, false);
    }
    catch (Exception ex)
    {
        MessageBox.Show(ex.Message);
        continue;
    }
}