I did this by hand and got 
$$ coker(f)=(R/J_{xv}\oplus R/J_{yv})/(xv,yv)$$, 
where $J_t$ is the ideal generated by all quadratic monomials except for $t$.

(Sorry for the garbled version of this I briefly posted earlier.)

<b>Edit:</b>  Graham Leuschke has helped me realize that I failed to mod out by the images of $(1,0)$ and $(0,1)$, so one should also mod out $(v,x)$ and $(y,z)$.