Is it possible for a family member on Apple plan to mark your device as lost?

Hi, yes according to the find my article a family member could put your device in lost mode with a few pieces of information.

Mark a device as lost in Find My on Mac:

https://support.apple.com/guide/findmy-mac/mark-a-device-as-lost-fmm0e3934777/mac

"Family Sharing: If you’re trying to mark a family member’s device as lost and the device doesn’t have a passcode set or is a Mac, you must enter that person’s Apple ID password on this device."

I hope this helped :D

If your family member opens the Find My app, selects 'Me' at the bottom.. scroll all the way down and there is an option to 'Help a Friend'. This will open iCloud login with Safari and let you log in with your Apple ID

OR

You can just sign into your iCloud account on any computer..