I don't know how many, but name scripted shows in which the theme songs were sung by a cast member or cast members.
Waylon Jennings sang the theme from
The Dukes of Hazzard.
Bobby Sherman sang the theme for the show
Getting Together:
Stretching things a bit:
Julee Cruise sang
Falling, the
Twin Peaks theme song, on a CD, on TV shows, and in a
Twin Peaks episode, (as "Roadhouse Singer"), but the actual theme song is is an instrumental version. Based on the wording of the question, this should qualify.
Bobby Sherman/ Here Come the Brides:
https://en.wikipedia.org/wiki/Seattle_(song)#Performances