No, space is not a property of something that moves.
You may have been misled by relativity textbooks which talk about reference frames as if they were part of some underlying physical reality. They are not. A reference frame is simply a way in which we can assign time and space co-ordinates to everything and every event in the universe.
In special relativity we often assign the origin of the reference frame to a particular object (let's call this A) at a particular point in time, and keep the origin of the three spatial co-ordinates fixed on this object. We can then measure distances and times of other events from that origin using light (or using measuring rods and clock that have been calibrated against light). But there is nothing special about A or the reference frame that we have "attached" to it - there are many, many other equally valid reference frames. And one of the two underlying assumptions or "postulates" of special relativity is that light has the same speed $c$ in every reference frame.
Another object B will define a different reference frame from A's. If B is stationary with respect to A then we can convert from the co-ordinates in A's reference frame to the co-ordinates in B's reference frame by just adding or subtracting a fixed amount from each of the three spatial co-ordinates. But if B is moving relative to A then we need some way of converting between co-ordinates in A's reference frame and co-ordinate's in B's reference frame. The constraints placed on this conversion or "transformation" are that the speed of light must be the same in all reference frames, and the laws of physics must not depend on which reference frame we happen to have chosen - because the reference frames are a useful convention for describing reality, but are not part of reality itself. Applying these constraints leads to the Lorentz transformation.
Special relativity replaced the earlier idea of the luminiferous aether, which was a like a reference frame, but was part of reality itself. Only objects that were stationary with respect to the aether would measure the speed of light as $c$ - other objects that were moving with respect to the aether would measure a different speed of light. But experiments that accurately measured the speed of light in the laboratory, such as the Michelson-Morley experiment, failed to detect any evidence that the Earth was moving through the aether. One attempt to patch up the luminiferous aether theory to accommodate these results was to assume that massive objects such as the Earth dragged some portion of the aether with them. But special relativity provided a much simpler and more consistent explanation of the experimental results, and did away completely with the need to have a privileged reference frame that was part of reality.
Special relativity and its extension in general relativity have been enormously successful in explaining many experimental results and observations about the universe on large scales (including gravitational waves, for example). Any alternative theory would have to be based on some very strong experimental evidence that SR and GR are incorrect, in the same way as SR and GR are based on experimental evidence that is inconsistent with previous theories. We do not have any such experimental evidence to discredit SR and GR or support any alternative.