As noted in the comments to this answer, the world "vacuum" has more than one meaning throughout Physics. In this answer, I'm taking the word to mean, in layman's terms, "the absence of matter". In technical terms, I mean "the stress-energy tensor vanishes", which is the classical notion of vacuum in General Relativity. I'm also assuming the cosmological constant to be zero (I take it to be a form of matter), but non-vanishing cosmological constants are already covered on John Rennie's answer. Finally, as mentioned in the comments, this answer interpreted the question as "suppose our Universe never had any matter" instead of "suppose all matter in our Universe suddenly vanished", but I am keeping it up because I believe it still adds useful information to the post.
That being said, within General Relativity, the absence of matter does not imply spacetime to be flat. For example, gravitational waves are a phenomenon that travels in the absence of matter. While they can be generated by the gravitational influence of massive bodies, they travel through vacuum and are an excellent example of how there can be gravitational effects in vacuum. The reason being that the gravitational field is not created by matter, but rather only interacts with matter.
Let me now turn to your question on whether spacetime would be curved in the absence of matter. Firstly, this is not a good approximation to our Universe. Physics we know depends heavily on the fact that there is matter, so there really is no way of knowing how the Universe would be in the absence of matter. For example, even the type of matter we consider when modelling the Universe changes conclusions such as if there ever was a Big Bang. We know there as a Big Bang due to experimental data, but that data includes the fact that there is matter. In other words, it is not trivial (and perhaps not even possible) to consider the effects of how gravity would be in the absence of matter. Matter matters.
As for how can spacetime be not flat in vacuum regions, or even in the hypothetical situation of a matterless Universe, that is due to the fact that the gravitational field also interacts with itself. In layman terms (this should be read with care), the energy of the gravitational field also interacts with the gravitational field, making it even stronger, which makes it interact more, which makes it stronger, and so on and so forth. These effects are one of the reasons why gravity is more difficult to study than, for example, electromagnetism. If in the beginning of time in that hypothetical Universe the gravitational field was curved for whatever reason, it likely will remain curved, possibly in different ways. Whether it would be curved or why it would be curved we can't know, for we can't predict this sort of information. Notice that Physics relies on taking some assumptions and computing consequences from them. If you take out all matter, you took out all our data together and we have nearly none assumptions, so we can't give you many predictions.
In summary, General Relativity allows spacetime to be curved in vacuum. That happens due to the fact that the gravitational field may interact with itself. On a vacuum Universe, if spacetime was ever curved, it would never be flat. If it were ever flat, it would remain flat. We can't really say anything about the "inherent curvature" of our Universe, for there is no way to consider how our Universe would be without matter. Matter matters.
