Create a js function that can be triggered by a click on a button or any other medium.
In Delphi you need to register an extension in the Browser.
Ex.
type
  TCustomRenderProcessHandler = class(TCefRenderProcessHandlerOwn)
  protected
    procedure OnWebKitInitialized; override;
  end;
  TExtensao = class
    class procedure click(const data: string);
  end;
For class TExtensao note that we have a procedure called click, this will be the contact with the js function, we are creating nothing more or less than a System between Delphi and Browser.
Procedure click:
class procedure TExtensao.click(const data: string);
var
  vMensagem : ICefProcessMessage;
begin
  // Registra o Evento Click para a Extensão que criamos anteriormente
  vMensagem := TCefProcessMessageRef.New('click');
  vMensagem.ArgumentList.SetString(0, data);
  TCefv8ContextRef.Current.Browser.SendProcessMessage(PID_BROWSER, vMensagem);
end;
Now, we need to inform the system that on startup it captures the events of the Browser, then in the boot section add:
initialization 
...     
  // Inicializa virtualmente os Procedimentos externos do Navegador que no caso vai registrar o TExtension}
  CefRenderProcessHandler  := TCustomRenderProcessHandler.Create;
  CefBrowserProcessHandler := TCefBrowserProcessHandlerOwn.Create;
The component TChromium has an event called ProcessMessageReceived, here every time the Istener click is triggered in the Navigator informs TCustomRenderProcessHandler that we registered earlier that a new message has arrived.
You can filter events through the parameter message. something like:
if message.Name = 'click' then
  message.ArgumentList.GetString(0)
As his name says message.ArgumentList is a list of arguments sent by the Navigator, and at position 0 contains the result of the js function that was created to pick the values of the desired fields.
Edit.
Missing implementation of the app Registration:
{ TCustomRenderProcessHandler }
procedure TCustomRenderProcessHandler.OnWebKitInitialized;
begin
  // Registra uma Extensão chamada "app" para o evento "click"
  TCefRTTIExtension.Register('app', TExtensao);
end;
							
							
						 
I already answered something similar, but there was no iteration with the author of the question: https://answall.com/questions/373648/como-pega-os-valores-dos-id-ao-click-do-bot%C3%a3o-html-no-tchromium-cef4delphi
– Junior Moreira