Yes, there are stable L1 and L2 Halo orbits in the Sun-Earth system.
When I say stable, I'm using the definition of "stable" that follows from examining the stability indices (eigenvalues) of the monodromy matrix. We describe the Earth-Moon NRHOs as "stable" because they represent a subset of the Halo family where both stability indices have a magnitude less than unity, or at least a magnitude that is not-much-larger than unity.
I can't find a more recent source, but we can at the very least get an answer from the master herself, Professor Howell. Her 1983 PhD dissertation, "Three-Dimensional, Periodic Halo Orbits in the Restricted Three-Body Problem," discusses families of orbits for the collinear libration points for a wide ranges of mass ratios. I'll quote her abstract here:
We found that families containing stable three-dimensional orbits exist near the two exterior libration points for all values of mu, and that the size of these orbits increases with mu. Near the interior libration point, stable orbits exist only for mu < .0573.
Mu being the mass ratio. The Sun-Earth mass ratio is 3e-6. Later, in Chapter II, section B, page 26, she says "We found that halo orbits exist at all values of mu. In most cases, stable halos also exist."
Later in this section she describes the stability indices for the L1 halos. There is a range of stable halos at L1 for the Sun-Earth mass ratio. Also, as the mass ratio decreases, the stable zone gets closer and closer to the primary (Earth). She shows a plot of the stability indices for different mass ratios, and it is clear that there is a stable zone for the Sun-Earth mass ratio.
She goes on to describe the L2 halo family. She says that stable halos exist at L2 or all values of mu tested, but I'm not sure if this includes the Sun-Earth mu. I've only given the dissertation a cursory read, but it looks like there are multiple regions for which the stability indices have magnitude less than unity. In particular, there is a stable zone as you get closer to the primary, but she says "As is the case with the L1 family, they become near-collision, almost rectilinear orbits and were not initially considered." She says that as mu gets smaller, so does the zone of stable orbits. She also shows a plot of the stability indices for L2 halos for a range of mass ratios, but the Sun-Earth mass ratio is not present in this plot. I'm not sure is this means that they don't exist, or, given her note about near-collision orbits, that they were too close to the Earth to warrant an appearance in the plot.
Regarding the Moon's effect on these orbits' stability, I'm not too sure either. It would really depend on how close the Moon gets to the Sun-Earth L1 and L2 halos. We know that we can maintain the almost-stable behavior for Earth-Moon NRHOs when we transition from the CRTBP model to the ephemeris model. So, if the Moon's gravitational influence on the Sun-Earth halos is smaller than the Sun's gravitational influence on the Earth-Moon halos, I would say that the Moon probably has very little effect on S-E halo stability.
However, there is one interesting note at the end of the Prof. Howell's dissertation abstract: "We also found that small amounts of eccentricity in the primary orbits apparently destabilize the periodic orbits of the infinitesimal body." The infinitesimal body is of course the spacecraft. I haven't read enough of the dissertation to know what range of eccentricities she studied, and how large this destabilizing effect is. Remember that even our "stable" Earth-Moon NRHOs are actually a little bit unstable, but the whole point is that they are so close to truly stable that we can perform stationkeeping with minimal fuel.