If they've kept their software updated they should not have any trouble
opening a Word file saved in either .doc or .docx format. However,
attachments can get butchered in the email process. If they *must* have a
Word file take the following steps:
1- Always save using file name extensions - Tick the checkbox in the Save As
dialog.
2- Stuff/Zip the file & attach it rather than the bare Word document.
3- Make sure your email app is properly set for encoding attachments
(depends on the app).
IMHO, it's better to send a PDF rather than a Word document. There's no
guarantee that a Word file will display/print as intended even if it arrives
unscathed. There are other factors that can prevent it from presenting
exactly as you designed it... and there's usually no valid reason for a
'headhunter' to have an editable copy of your resume. If absolutely
necessary, send both the Word file & a PDF.