Max,
Yes, you can certainly make the trip from Chamonix to Coermayeur, and then return. For that trip, you'll be using three different cable cars (first to Aguille du Midi which has two segments as I recall, then the Panoramic Mont Blanc over the glacier and then the new car down to Coermayeur).
I've been as far as Pointe Helbronner, but didn't make it down to Coermayeur. You didn't mention when your trip will be taking place. Be sure to check the schedules as the Panoramic Mont Blanc only operates from about May to September.
One other important point to mention is that if you're susceptible to "altitude-related issues", it would be prudent to limit your time at the higher elevations, especially the Aguille du Midi which is at about 12,600 feet / 3840 M. If you experience difficulties, head to a lower elevation as soon as possible.