Can I replace the wax gasket without lifting the whole toilet?
Thinking you are looking to just replace the tank gaskets, here is the scoop.  There is no wax seal between the tank and bowl, there is usually a neoprene or rubber gasket that needs to be the right size and shape for the model of toilet you have. You should also replace the mounting bolts and the rubber gaskets around them also at the same time.  As far as trying to lift one side at a time, i guess that could be done if both bolts were removed and you tilt the tank enough to clearly see the tank drain hole and properly insert the new seal. Of course, it is better to install the seal into the tank and lower it square onto the bowl opening then install the new bolts and gaskets.  The real issue is to be sure the new main gasket is seated squarely so it will seal properly. Be careful tightening the tank bolts. They need to be firm to the point where there are no leaks and the tank is level, but not too tight as over tightening these bolts can crack the tank or mounting flange on the bowl.

Maybe you can find a strong friend to give you a hand with this tricky job. It isn't a difficult task, but might be frustrating if you can't remove the tank, clean the drain hole and install the new gasket at a more convenient working angle.