You need to use a frequency that the atmosphere is transparent to. I would suggest using 4 GHz microwaves, the same as is used in your kitchen appliance.
You would bath the world with enough microwaves to raise the temperature of the typical critter by about 10 degrees C. This would leave the corpse in roughly the same condition as dying from a fever would have. Critters that were under dense forest cover would have a higher survival rate. Critters in the water would be able to dissipate the heat.
This is still a not-trivial amount of energy. If you are trying to heat up an average of 30 cm thick layer of water by 10 C, you are spending 600 kCal/square foot or about 2.4 MJ or about 2/3 of a kWh/ft2.
Expanding that, about 7 kWh/m2 = 7 million kWh/km2 Earth is about 500 million square km so 3,500 trillion kWh. And that assumes efficient coupling.
This seems very wasteful. Food microwaves, after all, were designed to cook. It may be possible to come up with a frequency that hits a resonance in some protein. Imagine something that disrupted hemoglobin. Zzzzot! everything with a red blood cell falls over dead.
If you are willing to criss cross the planet within the atmosphere, you could do everyone in with sound. kWh per square meter will make a lot of noise. (Hifi speakers are terribly inefficient at converting energy to noise. The designers care about fidelity. even 1 watt of acoustic energy is LOUD. Suppose you generated infra-sound, say at 10 Hz, but at an amplitude that had peak over pressure measured in pounds per square inch. As a side effect it would treat most buildings like a bad earthquake.
However, overall, chemistry is more suitable. The ideal chemical is one that requires some time after exposure to be effective. This means that people won't run and hide when others start dying. Something that interferes with a precursor chemical for nerve transmitters may do. Circle the planet in a polar orbit, and kick the pellets out of the ship at a speed they are near motionless relative to the surface. This means they hit the atmosphere fairly slowly so they don't burn up.
A way to convert all or much of the carbon dioxide in the atmosphere to carbon monoxide would do the trick. CO has a residence time in the atmosphere of between 2 months and 5 years. (Estimates vary...) So discovery a week after the event would leave a big fingerprint. WHO exposure limits are 10 ppm for a shift, and 100 ppm for 15 minutes. This is sufficient to block 2% of the hemoglobin. Residence time in the blood is about 5 hours, so the 100ppm should eventually stabilize at around 40%. It may be possible to excite CO with the right frequency to give it the energy to reconvert it back to CO2.
Diseases could work too. Spanish flu was the most recent pandemic with high mortality. While somewhere between 50 and 100 million people died, overall mortality was a small percentage of the world's population. Perhaps another disease that kills with a Cytokine storm. (Flu was scary. A person could go to bed healthy and wake up dead.)
Diseases seldom are deadly to multiple species. Rabies is an exception, and may be a good candidate:
- Most mammals are susceptible.
- You can get it from inhalation. (Bat researchers now get vaccinated against it.)
- It has a fairly long incubation time.
No disease is sudden enough to not leave records. Anything lethal enough that people can't write stuff down is too lethal to spread effectively.
Finally, if you truly want to leave a mystery, and have Startrek transporter technology, you circle the planet, stealthed, and beam people up, dropping them on the back side of the moon. Don't worry about maintaining the pattern buffers... At 100 people per second, 7 billion people would take 7 million seconds, or about 3 months. 10,000 people per second brings it down to 70,000 seconds or about a day. This is about the same time as a polar orbit will cover the earth.
Mind you, if you have transporter technology, tune it to a particular molecule, and just remove nerve transmitters. No one is going anywhere if their acetylcholine is missing.