If you have a sentence that is referring to several people who share the same title, does the title become plural?
The position they are being appointed to is titled Associate so this would be correct:
"We are excited to announce the promotion of John, Bob, and Joe to the position of Associate." (Leave out your comma before to.)
Alternatively, you could talk about their individual positions:
"We are excited to announce the promotion of John, Bob, and Joe to their new positions as Associates."
Each one is an Associate, and they are all Associates. Each one has the position of Associate.
GArthurBrown's answer is entirely correct, though I would like to add that titles are pluralized when they are used before the names in question. For example:
"We are pleased to introduce professors Einstein and Newton."
"Doctors Kahneman and Anderson are the foremost modern psychologists."
"The suspects were taken into custody by officers Walker and Trivett"