with() and within() functions inside lapply() not seeing outside of its environment?

I wouldn't call it a bug, but it's a documented limitation, if you know 
how to read it.  As documented, the expression is evaluated with the 
caller's environment as the parent environment.  But here the caller is 
some code in lapply, not your function f.  x is not found there.

I think this modification works, and is maybe the simplest way to get it 
to work:

f <- function(x){
   y <- list(y1=list())
   mywithin <- function(...) within(...)
   y <- lapply(y, mywithin, {z<-x})
   y
}

The idea here is that the calling frame of f is the environment of 
mywithin(), so x is found there.