Fillan Grady already gave the examples of orbiting a Lagrangian point, but based on your response to that, and subsequent edit, it seems you want a more traditional Earth orbit, rather than a Lagrangian "cheat". But I will point out that these points are valid ways to orbit the earth, and there is a reason they are so useful.
Bottom-line: If you discount the L-points, it's not very likely, and certainly not in a low orbit. Every satellite orbit must follow a great circle, and therefore, must cross the equator at two points. This is true, even for polar orbits. Furthermore, these crossing points will remain fixed, relative to the celestial sphere (unless you invoke some sort of gravitational precession), while the Moon revolves around the Earth. That means that at some point, the Moon will almost certainly be opposite of the Earth during this crossing.
What you have to do is find a polar orbit high enough (and with the correct period and phase, relative to the Moon's orbit) that the Earth is small enough to not occult the moon during any of these crossings, ever. Consider a satellite in geosynchronous orbit (24 hour period). A back of the envelope calculation says that the earth will appear to have an angular diameter of about 20 degrees. The angular portion of the Moon's orbit which this hides, on the opposite side the Earth, will be even greater than this. However, assuming a 27 day circular orbit, the Moon only covers about 13 degrees of its orbit per day, so if the Moon is just about to go behind the Earth during one orbit, it will not have come out the other side by the time your satellite returns to the same point 24 hours later.
Now, a geosynchronous polar orbit would already take quite a bit of energy to achieve, and it isn't even close to providing what you would need. You'd have to go quite a bit higher, and it would almost certainly be cheaper to launch to L1, or to use network of multiple satellites on lower-altitude, lower-inclination orbits. But you've said that you want a low-altitude orbit -- that will make the Earth appear bigger, which will increase the time the Moon is behind it, while simultaneously increasing the frequency with which the satellite is in this shadow.