--- --- 
type
  TScrollingPaintBox = class(TScrollingWinControl)
  private
    FCanvas: TCanvas;
  public
    constructor Create(aOwner: TComponent); override;
    destructor Destroy; override;
    property Canvas: TCanvas read FCanvas;
  end;

constructor TScrollingPaintBox.Create(AOwner: TComponent);
begin
  inherited Create(AOwner);
  FCanvas := TControlCanvas.Create;
  TControlCanvas(FCanvas).Control := Self;
end;

destructor TScrollingPaintBox.Destroy;
begin
  FCanvas.Free;
  inherited Destroy;
end;

A TControlCanvas is important because it creates a DC (device context) that belongs to the HWND of the control. Also, override is important on your constructor and destructor to ensure that they are actually called.