If mass curves spacetime, why do planets in a vacuum follow curved paths?

The curvature of spacetime can be separated mathematically into two components, Ricci curvature and Weyl curvature. They are locally independent, but their joint variation over spacetime is constrained by mathematical relations (the second Bianchi identity).

General relativity says that the Ricci curvature is determined by the local matter density (stress-energy), but there is no direct constraint on the Weyl curvature.

So, in vacuum regions (Schwarzschild field, gravitational waves, etc.), the Ricci curvature is zero while the Weyl curvature can be nonzero. The physical value of the Weyl curvature is determined by the mathematical curvature relations and the boundary conditions.

The Weyl curvature represents the propagating degrees of freedom of the gravitational field, which can exist without matter. This spreads the influence of gravity beyond the immediate location of matter, but does not represent action at a distance because it still acts causally (limited by the speed of light).

Typically we solve for the gravitational field of the Sun as a steady state, which makes it seem like a global result that appears all at once. However, if we pose an initial-value problem containing a central mass (thus determining Ricci curvature) with a different initial configuration of Weyl curvature, the "extra" Weyl curvature would break up into gravitational waves and ultimately disperse to large distances, leaving the steady-state (Schwarzschild) solution.

That is, roughly speaking, the Weyl curvature is indirectly determined by matter, as the effect of Ricci curvature on Weyl curvature propagates outward at the speed of light.

The curvature extends far away from the mass creating it, becoming progressively gentler as the distance increases, and it never completely goes away. The same curvature becomes stronger as the distance decreases, making all the effects of gravity more powerful the closer you get to the mass.

These are the characteristics of a field which extends throughout space, and upon which things like matter and electrical charge can act. For all known fields like this, there is a finite speed with which disturbances can travel through it and be felt later by distant objects and that speed is c, the speed of light.

This is not a perfect analogy, but imagine a rubber sheet, held flat in a frame. Now pinch as small circular region in the rubber (eg, push the sheet through a small ring). The small pinched region causes the rubber to stretch, with greater stretching near to the ring, becoming progressively less away from it. The pinched ring is analogous to a gravitating mass, and the stretching of the sheet is analogous to the scaling distortion of a map of space surrounding a gravitating body (bearing in mind that what we mean by curvature mathematically is actually scaling distortions of maps, not something curved in the usual sense of the word).

Of course it is a bit more complicated if one considers spacetime rather than just space, but one cannot distort space without also distorting spacetime.