If you activated the vacation responder but your email signature is not showing up on your vacation responder text, because by default, Outlook does not automatically insert your signature into the vacation responder text box.
To fix this issue, Outlook (Classic)
Go to File > Automatic Replies.
Select Send automatic replies.
Important: Outlook doesn't add your signature automatically here. Copy and paste your signature directly into the text box.
Use the formatting bar (above the text) to make sure it looks correct.
Click the Outside My Organization tab and check the box to send replies to Anyone.
Click OK.
New Outlook:
Open Settings: Click the Gear Icon ⚙️ at the top right.
Select Mail > Automatic Replies.
Switch the toggle to "Turn on automatic replies."
Add Your Signature: The New Outlook does not add your signature automatically.
Manually copy and paste your signature directly into the message box