Nebo zjednodušeně nějak takto:
uses
  fphttpclient;
procedure Download(AURL: string);
var
   AClient: TFPHTTPClient;
   AStream: TMemoryStream;
   AFileName: string;
begin
  AClient:= TFPHTTPClient.Create(nil);
  AStream:= TMemoryStream.Create;
  try
    AClient.Get(AURL, AStream);
    AFileName:= IncludeTrailingPathDelimiter(ExtractFilePath(Application.ExeName)) + ExtractFileName(AURL);
    AStream.SaveToFile(AFileName);
  finally
    AClient.Free;
    AStream.Free;
  end;
end;
...
Download('https://i.iinfo.cz/images/214/rust-1-prev.jpg');
předpokladem je, že v systému existuje openssl knihovna