I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.
Then by that, can we say that
It can be used, but it's a bad piece notation, just as $y'$ is. I have two reasons for saying that.
$y'$ doesn't indicate with respect to which variable you differentiate. So for example, if $y=t^2$ and $t=e^x$, what should $y'$ denote? Is it $dy/dt$ or $dy/dx$? I see students getting confused by this when they try to derive with the chain rule using the prime notation.
Even if you object and say: in my context $y'$ will always denote derivative with respect to $x$, I still consider it a bad notation since it's the same notation we use for $f'$. Hence it suggest to students that $y$ and $f$ are the same type of objects.
The only justification I see for using it is laziness and tradition.