Yes, it is possible.
- Radius of Earth:$6,378 \mathrm{km}$
- Radius of the Moon: $1,737\mathrm{km}$
- Semimajor Axis of the Moon's orbit: $384,000 \mathrm{km}$
- Minimum distance from Earth to Mars: $54,600,000 \mathrm{km}$
Since the maximum difference in distance between the Earth and the moon is roughly 0.7% the minimum distance between Earth and Mars, you can effectively ignore the tiny change in apparent size due to the Moon being 384,000 km closer when between Earth and Mars, and can compare their diameters directly.
As a result, the apparent size of the Moon is always smaller than the apparent size of the Earth when viewed from Mars. The Moon cannot simultaneously block line-of-sight from all points on the Earth's surface to Mars.
And any communication link from a point on Earth to Mars or back will include some kind of relay in the design, because roughly 50% of the time, Mars will be below the horizon from said point on Earth.