React: Can I check if a state exists before rendering it

Sure, use a ternary:

render() {
  return (
    this.state.name ? <NavItem>{this.state.name}</NavItem> : null
  );
}

or even shorter

render() {
  return (
    this.state.name && <NavItem>{this.state.name}</NavItem>
  );
}

Sure you can:

let userNavItem
if (this.state.name !== undefined) {
  userNavItem = <NavItem eventKey={4} href="#">{this.state.name}</NavItem>
} else {
  userNavItem = null
}

Now you can use userNavItem on your navbar component, and it will render only if this.state.name is defined.