Typically we use a combobox or listbox to allow to the final user select and appy a VCL Style, Today I will show you a new way using the system menu of the form.

First you need to use the GetSystemMenu WinApi function  to get a  handle to the system menu of the form. Then using the AppendMenu or the InsertMenuItem methods you can customize the system menu, from here you must store the identifier of the new menu item added and finally process the WM_SYSCOMMAND message to launch an action.

Check the next commented code

uses
  System.Rtti,
  System.Classes,
  System.Generics.Collections,
  WinApi.Windows,
  WinApi.Messages,
  Vcl.Themes,
  Vcl.Styles,
  Vcl.Forms;

type
  TMethodInfo=class;

  TProcCallback = reference to procedure(Info : TMethodInfo);
  TMethodInfo=class
   Value1 : TValue;
   Value2 : TValue;
   Method : TProcCallback;
  end;
  TVclStylesSystemMenu=class(TComponent)
  strict private
    FVCLStylesMenu : HMenu;
    FOrgWndProc: TWndMethod;
    FForm : TForm;
    FMethodsDict : TObjectDictionary<NativeUInt, TMethodInfo>;
    procedure CreateMenus;
    procedure DeleteMenus;
    procedure CreateMenuStyles;
    procedure WndProc(var Message: TMessage);
  public
    constructor Create(AOwner: TForm); reintroduce;
    destructor Destroy; override;
  end;

implementation

uses
  Vcl.Controls,
  System.SysUtils;

const
 VCLStylesMenu=WM_USER + 666;

//Add a new Menu Item
function InsertMenuHelper(hMenu: HMENU; uPosition: UINT; uIDNewItem: UINT_PTR; lpNewItem, IconName: LPCWSTR) : BOOL;
var
  LMenuItem : TMenuItemInfo;
begin
  ZeroMemory(@LMenuItem, SizeOf(TMenuItemInfo));
  LMenuItem.cbSize := SizeOf(TMenuItemInfo);
  LMenuItem.fMask  := MIIM_FTYPE or MIIM_ID or MIIM_BITMAP or MIIM_STRING;
  LMenuItem.fType  := MFT_STRING;
  LMenuItem.wID    := uIDNewItem;
  LMenuItem.dwTypeData := lpNewItem;
  Result:=InsertMenuItem(hMenu, uPosition, True, LMenuItem);
end;

//Add a new separator
procedure AddMenuSeparatorHelper(hMenu : HMENU; var MenuIndex : Integer);
var
  LMenuInfo    : TMenuItemInfo;
  Buffer       : array [0..79] of char;
begin
  ZeroMemory(@LMenuInfo, SizeOf(TMenuItemInfo));
  LMenuInfo.cbSize := sizeof(LMenuInfo);
  LMenuInfo.fMask  := MIIM_TYPE;
  LMenuInfo.dwTypeData := Buffer;
  LMenuInfo.cch := SizeOf(Buffer);
  if GetMenuItemInfo(hMenu, MenuIndex-1, True, LMenuInfo) then
  begin
    if (LMenuInfo.fType and MFT_SEPARATOR) = MFT_SEPARATOR then
    else
    begin
      InsertMenu(hMenu, MenuIndex, MF_BYPOSITION or MF_SEPARATOR, 0, nil);
      inc(MenuIndex);
    end;
  end;
end;

{ TVclStylesSystemMenu }

constructor TVclStylesSystemMenu.Create(AOwner: TForm);
begin
  inherited Create(AOwner);
  //Get an instance to the form
  FForm:=AOwner;
  //Init the collection to store the menu ids and callbacks
  FMethodsDict:=TObjectDictionary<NativeUInt, TMethodInfo>.Create([doOwnsValues]);
  //store the original WndProc
  FOrgWndProc := FForm.WindowProc;
  //replace the WndProc of the form 
  FForm.WindowProc := WndProc;
  //Modify the system menu
  CreateMenus;
end;

destructor TVclStylesSystemMenu.Destroy;
begin
  DeleteMenus;
  FForm.WindowProc := FOrgWndProc;
  FMethodsDict.Free;
  inherited;
end;

procedure TVclStylesSystemMenu.CreateMenus;
begin
  CreateMenuStyles;
end;

procedure TVclStylesSystemMenu.DeleteMenus;
begin
   if IsMenu(FVCLStylesMenu) then
   while GetMenuItemCount(FVCLStylesMenu)>0 do
     DeleteMenu(FVCLStylesMenu, 0, MF_BYPOSITION);

   FMethodsDict.Clear;
end;

procedure TVclStylesSystemMenu.CreateMenuStyles;
var
 LSysMenu : HMenu;
 LMenuItem: TMenuItemInfo;
 s : string;
 uIDNewItem, LSubMenuIndex : Integer;
 LMethodInfo : TMethodInfo;
begin
  LSysMenu := GetSystemMenu(FForm.Handle, False);

  LSubMenuIndex:=GetMenuItemCount(LSysMenu);
  AddMenuSeparatorHelper(LSysMenu,  LSubMenuIndex);

  FVCLStylesMenu   := CreatePopupMenu();
  s:='VCL Styles';

  uIDNewItem := VCLStylesMenu;
  ZeroMemory(@LMenuItem, SizeOf(TMenuItemInfo));
  LMenuItem.cbSize := SizeOf(TMenuItemInfo);
  LMenuItem.fMask  := MIIM_SUBMENU or MIIM_FTYPE or  MIIM_ID or MIIM_BITMAP or MIIM_STRING;
  LMenuItem.fType  := MFT_STRING;
  LMenuItem.wID    := VCLStylesMenu;
  LMenuItem.hSubMenu := FVCLStylesMenu;
  LMenuItem.dwTypeData := PWideChar(s);
  LMenuItem.cch := Length(s);
  //Add the new menu item to the system menu
  InsertMenuItem(LSysMenu, GetMenuItemCount(LSysMenu), True, LMenuItem);
  inc(uIDNewItem);
  LSubMenuIndex:=0;

  //Iterate over the registered styles and create a new menu entry for each style 
  for s in TStyleManager.StyleNames do
  begin
    InsertMenuHelper(FVCLStylesMenu, LSubMenuIndex, uIDNewItem,  PChar(s), nil);
    if SameText(TStyleManager.ActiveStyle.Name, s) then
      CheckMenuItem(FVCLStylesMenu, LSubMenuIndex, MF_BYPOSITION or MF_CHECKED);
    inc(LSubMenuIndex);
    inc(uIDNewItem);
    LMethodInfo:=TMethodInfo.Create;
    LMethodInfo.Value1:=s;
    //set the method to execute when the item is clicked
    LMethodInfo.Method:=procedure(Info : TMethodInfo)
                        begin
                          TStyleManager.SetStyle(Info.Value1.AsString);
                        end;
    //register the menu id and the callback function.
    FMethodsDict.Add(uIDNewItem-1, LMethodInfo);
  end;
end;

procedure TVclStylesSystemMenu.WndProc(var Message: TMessage);
var
  LVerb : NativeUInt;
begin
  case Message.Msg of
    //Detect when the window handle is recreated
    CM_RECREATEWND: begin
                      DeleteMenus;
                      FOrgWndProc(Message);
                      CreateMenus;
                    end;
    //Track the system menu calls
    WM_SYSCOMMAND : begin
                     if FMethodsDict.ContainsKey(TWMSysCommand(Message).CmdType) then
                     begin
                      LVerb:=TWMSysCommand(Message).CmdType;
                      FMethodsDict.Items[LVerb].Method(FMethodsDict.Items[LVerb]);
                     end
                     else
                      FOrgWndProc(Message);
                    end
  else
    FOrgWndProc(Message);
  end;
end;

end.

And this the result

To use this class, only you need create an new instance passing a reference to the form.

procedure TForm1.FormCreate(Sender: TObject);
begin
  TVclStylesSystemMenu.Create(Self);
end;

You can check the full source code here.