Is the Cosmological Constant locally zero?

No. The cosmological constant is indeed a constant. It is what it is - both locally and globally. The fact that we don't notice the effects of a non-zero cosmological constant at small scales is certainly because the cosmological constant is very small. In principle, the non-zero cosmological constant does contribute to the expansion of the solar system as well. But certainly, the effect is ridiculously negligible at the scale of the solar system.

As I have stated in my answer to a similar question here the cosmological model is not about bound states, and even galaxies are bound states by the gravitational attraction. Certainly bound states are presumed to overcome any distortions of space time; if it were not so all our cosmological observations of velocities which depend on identifying the Doppler shifted spectra of known elements come into doubt. Also the standard Big Bang model becomes nonsense: if space between bound particles expanded, with the cosmological constant , we would not be able to measure the expansion.

At the local level in order of strength, a proton is bound by the strong force, the hydrogen atom by the electromagnetic, the earth to the sun by gravitation. A look at the relative contribution to the space curvature by Λ (equation 1 here ) shows that Λ is order of 10^20 smaller than the contribution of the gravitational constant.

At the quantum level of the interactions binding atoms and molecules, Λ would appear as an extra dispersing potential, modifying the quantum levels measured . The smallness of the number locally ( look at the answer by Ben Crowell here) assures that within the widths of the bound states the effect is infinitesimal. It is similar to not considering the gravitational attraction between electron and proton when calculating the bound states of hydrogen. For the reason that the effect of changes in gravitational attraction are not seen , see this answer by Ben Crowel, .

So it is only at cosmological levels that the expansion driven by Λ can have a measurable effect, not at a local one.

The cosmological constant or dark energy is present everywhere and at all scales according to theory. (Experimentally we don't know that it is, though it certainly doesn't clump very much if at all.) But it doesn't, even in theory, cause a time variation of orbital parameters.

The easiest way to understand its effect is to think of it as matter with a uniform density of roughly $ρ_Λ approx -10^{-26}text{ kg}/text{m}^3$ (or around −10 yoctograms per m³ if that's easier to remember).* Its effect on orbits is therefore (by the shell theorem) roughly equivalent to a decrease of the central mass by $frac43 π r^3 ρ_Λ$ where $r$ is the radius of the orbit. Because this decrease doesn't vary with time, it doesn't have a time-varying effect on the orbit. It merely makes it a little smaller or a little slower than it would have been otherwise.

If $r = 1text{ AU}$, then the decrease is about $10^8text{ kg}$. To put this in perspective, the sun loses around $5times10^9text{ kg}/text{s}$ in the form of light, neutrinos, and solar wind. That continuous loss actually should in principle cause a time variation in orbital parameters. The cosmological constant doesn't increase the rate of the change; it merely shifts it in time by a fraction of a second.

The radius at which the sun's mass is entirely canceled, meaning there are in theory no solar orbits beyond that radius, works out to about $100text{ pc}$. In practice this is meaningless because it assumes that there are no other gravitating bodies within that distance of the sun.

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* The dark energy density is actually positive. The reason this "effective density" is negative is that the acceleration in GR is proportional to $ρ+3p$, with $p=-ρ$ for dark energy. In Newtonian gravity only $ρ$ gravitates, but you can simulate the GR acceleration by multiplying $ρ$ by $-2$.