This is an awkward sentence, but I believe it means "as well as she could *while* sneezing." So she was having trouble speaking because the pepper made her sneeze a lot.

This seems to make a bit more sense, so I'll try to explain. "There was certainly too much pepper in that soup!" Alice said to herself, as well as she could for sneezing." Here, a situation is described. The "as well as she could for sneezing" implies that she spoke that sentence _while_ sneezing. Furthermore, the sentence describes a condition as well, so as to describe the presented action in a broader way. - It is stated that she isn't speaking her sentence properly, as she speaks it mid-sneeze. - To be honest, I don't like the phrasing of that sentence all too much...

There is nothing "awkward" about the sentence. If anyone commenting "doesn't like" Lewis Carroll's English, then that's their problem. He was an incredible writer. His English is from another era, that's all. To modern ears, his form of English may sound slightly dated if you're not used to reading literary gems from the past and are not used to reading sophisticated English. To do something "as well as you can for XXXing" simply means that a secondary action is impeding you from properly carrying out the first. In this example, Alice's sneezes are impeding her from speaking properly, so she is merely trying to speak as well as she can.

Thank you very much

As well as she could WHILE sneezing would make more sense here to me.