According to LogMeIn Communities Forum, LogMeIn support said they are working on M1 compatibility.
Here's a workaround that does not require physical access to the machine:
- From the LogMeIn Desktop Client, login to the machine
- Once you connect, you'll only see a black screen, please leave this window open
- Go back to the client and launch the terminal
- In the terminal prompt, type caffeinate -u -t 2 and press enter.
- You'll then be able to view the login screen and access your Mac normally.
Apparently this works because caffeinate with these arguments tells the M1 Mac that there is a user present at the computer, so it turns on the video.