Could this woman be placed on trial in Arizona? If not, then what is the purpose of a law effectively banning abortion within the state?
No. She was out of state engaging in legal behavior in the state that she was in.
Arizona or any other state cannot "effectively outlaw abortions". Only the Supreme court could do that. If any state tries (like South Dakota did) it would get slapped down as unconstitutional, because abortion is LEGAL everywhere in the republic.
They may try to put roadblocks to access to abortion services, but ultimately, if a woman wants one she CAN get one. In any state.
The government charges and prosecutes Americans here for engaging activities that are legal in other countries.
The FEDERAL government. States can't