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By using the exponential class factorization theorem, I came up with $Y = \sum (x_i)^2$ to be the complete and sufficient statistics for $\sigma^2$ .

Using this sufficient statistic as a condition, would I need to get the MLE to derive the MVUE for $\sigma^2$ ?

I don't know how to proceed to get the MVUE.

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If $Y$ is complete sufficient, then $f(Y)$ is the unique (up-to null sets) UMVUE of its expectation (Lehmann-Scheffe). In particular, take $f(Y) = Y / n$ and check that $E[f(Y)] = \sigma^2$.

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