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Note that when getting a shipping email it doesn't actually have to mean it is shipped, but rather that the package is announced for shipping at the shipping provider. For US shipments, this usually is a shipment within the week (they start with domestic shipments first), but for international shipments it can take a week or two before it is actually shipped. If you see the package move in the tracking code, then it is a sign that it has actually been shipped. The reason why it takes a while is because they are a small company (I think they have two dedicated persons on shipping duty), and shipping out thousands of orders at once takes a while. This run was around 8000 PS4 and 6550 vita copies, iirc.
If you don't see the package move in the tracking code a few weeks after receiving it, then you should contact Limited Run.
And on social media, just yesterday they were still announcing early October as a shipping date, so it might still take a while.
I didn't receive a shipping email myself yet, but this is based on earlier experience with Limited Run.