It depends what distribution you start the process in. Yes if the initial distribution is
\Gamma(0) & \Gamma(-1) & \cdots & \Gamma(1-L) \\
\Gamma(1) & \ddots & \cdots & \Gamma(2-L) \\
\vdots & \vdots & \cdots & \vdots \\
\Gamma(L-1) & \Gamma(L-2) & \cdots & \Gamma(0)
Is it correct that due to the invertibility of the VAR into an MA and
observing that Xt is the sum of zero mean MVN random variables the
above is true, or is there a flaw somewhere in this.
First, causality (not invertibility) lets you write the model as a sum of the infinite past of $\epsilon$ terms. The above condition doesn't posit the existence of an infinitely long history of data, but if you wanted to go that way, then yes, that will work, too. It is because linear combinations of Normal vectors end up being Normal as well.