I'm looking for some help in breaking up this double summation into a more usable formula. Something where I can plug in values for yi, yj, and Vcij to solve for Vcm.

Double summation means you need to add the terms for every combination of i and j.

For example, if $V_C$ is a $3\times3$, then $V_{Cm} = y_1y_1V_{C1,1} + y_1y_2V_{C1,2} + y_1y_3V_{C1,3} + y_2y_1V_{C2,1} + y_2y_2V_{C2,2} + y_2y_3V_{C2,3} + y_3y_1V_{C3,1} + y_3y_2V_{C3,2} + y_3y_3V_{C3,3}$.

In some special cases you can simplify further, but generally you can't.

I'm looking for some help in breaking up this double summation into a more usable formula. Something where I can plug in values for yi, yj, and Vcij to solve for Vcm.

I don't know what you mean by "more usable". However, if you want to represent it in a nicer way to simplify some linear algebra computations then you can rewrite this by noting it is a quadratic form. Specifically, if $y$ is a column vector and $V$ is a matrix whose entries are $V_{ij}$, then the formula for $V_{Cm}$ simply reads: $V_{Cm} = y^T V y$. However, I strongly suspect this is where you got such a formula to begin.