He certainly did and his voice wasn't bad either. Here's his own version of Galveston, one of a number of hits he wrote for Glen Campbell.

 

https://www.youtube.com/watch?v=U_YK_L_Fr4Q

'Galveston' by JIMMY WEBB -from the 1972 album LETTERS- -Regarded by many critics as Jimmy's best solo album, "Letters" finds Webb working in a number of styles, but none quite so devastating as his simple, acoustic rendition of the hit made popular by Glen Campbell, 'Galveston'.