Is there a way to start a child process without fork(), using execvp() exclusively?
The pedantic answer to your question is no. The only system call that creates a new process is fork. The system call underlying execvp (called execve) loads a new program into an existing process, which is a different thing.
Some species of Unix have additional system calls besides fork (e.g. vfork, rfork, clone) that create a new process, but they are only small variations on fork itself, and none of them are part of the POSIX standard that specifies the functionality you can count on on anything that calls itself a Unix.
The slightly more helpful answer is that you might be looking for posix_spawn, which is a library routine wrapping fork and exec into a single operation, but I find it more troublesome to use that correctly than to write my own fork+exec subroutine. YMMV.