Does vacuum spacetime have an inherent curvature? I am a complete novice in physics beyond the high school level, so please excuse anything wrong in my question.
So I have recently read that according to General Relativity, the presence of mass in spacetime causes spacetime to become curved and that is how gravity arises.
Does vacuum spacetime have an inherent curvature? What I mean to say is that if we remove all kinds of matter and energy from the universe (somehow), and then try to determine the curvature of spacetime, will it be flat or will it be curved?
And if vacuum spacetime does have an inherent curvature, why or how does that curvature arise, given that nothing possessing energy or mass is present in the universe I have described above.
 A: 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.
A: The Einstein equation allows spacetime to have an inherent curvature, but this is an adjustable parameter. That is, general relativity does not predict what the inherent curvature would be, only that it could exist and could take any value. The only way we can tell whether spacetime has an inherent curvature or not is by observation.
One of the terms allowed in the Einstein equation is a cosmological constant, normally written as $\Lambda$. If $\Lambda$ is non-zero then spacetime has a scalar curvature (the Ricci scalar) given by:
$$ R = 4\Lambda $$
And this is exactly the sort of curvature that you are asking about because it is "built in" to spacetime and exists even in a universe completely empty of matter or energy.
In fact does appear that the universe could have a cosmological constant. When we observe the motion of supernovae in the universe it looks as if the universe is expanding faster than it should, and this could be due to a cosmological constant. The trouble is that it could also be due to a form of energy called dark energy, and at the moment we cannot tell which (if either) of these is the case. One way to tell would be to see if the effect changes with the age of the universe. A cosmological constant would be ... well ... constant, while dark energy could change with time. However at the moment our measurements are not precise enough to tell if the effect has changed over the age of the universe.
A: A spacetime without matter or energy is called a vacuum spacetime. There are flat vacuum spacetimes as well as vacuum spacetimes with curvature.
The reason for this is that, like most differential equations, different solutions can be obtained for different boundary conditions, even given the same sources.
So just like vacuum solutions for Maxwell’s equations include both no field solutions and plane wave solutions, similarly vacuum solutions in GR include flat spacetime, gravitational waves, and other curved spacetimes.
