Writing on someone's behalf

If the letter is authorized/cleared by your supervisor to go out its ok

I imagine different companies have different protocols.

Some may want you to sign your own name with a "for" before person's signature block.
Others may want you to do that "pp" thing (there are countless threads on that in here).

Ask the most senior administrative assistant you can find how it's handled in your company. She will know.

I am writing thank you notes for my elderly Mother after her husband passed away. Should I start the thank you note with ,On behalf of my Mother... and I also wonder how to sign the note, my name or my Mother's? Thank you.