This is in the U.S. I am a graduate student, and an undergraduate student who I mentored in an independent reading project has asked me to write her a letter of recommendation for an internship outside of the university.
For some context, I am the organizer of the program at my university through which undergrads apply to do these projects, and this program is advertised on the university website.
I haven't been able to find any information on who can use the letterhead on my university's website, thought I'm sure it's buried there somewhere. I was only able to one other question about this on here, and it was about an undergrad using the letterhead for a personal matter.