I have a Windows 10 desktop computer (with no SIM card) with a program installed that lets me send an SMS message when a certain event fires [...].
The program lets me choose my COM1 interface as the output interface for the message.
What physical hardware would I need to send an SMS from the desktop computer?
A GSM modem. In the '90s I had a mobile phone that could act as one over a serial port.
Today, to programmatically send SMS, most people will use a web service (such as Amazon SNS or Twilio MessageX) instead of dedicated hardware.