Does Lady Macbeth have the ability to decide fate?
It's impossible to know whether Shakespeare was himself a Christian—but he was definitely writing in a Christian culture to a Christian audience. Within such a context, fate is something that is predestined in accordance with God's will. It is in the nature of fate that humans cannot decide upon it.
As to whether is is possible for Lady Macbeth to change her fate or that of her husband, that gets into complex notions of predestination, free will, and future contingency. As humans with free will, we like to think that we are capable of making decisions. Such a belief is also necessary for us to believe in divine justice, as it seem unfair that we be rewarded or punished if we do not have free will to make choices resulting in those rewards or punishments.
In the play, there seem to be choices, though. First, because witches are evil, Macbeth could have refused to listen to them and Lady Macbeth could have tried to dissuade her husband from heeding their advice. Even if Macbeth was destined to be king, he could have striven to be a loyal subject to King Duncan and perhaps if he had been morally good, he might have been awarded the kingdom after all for his good deeds or military valor.
When Lady Macbeth receives the letter about the witches' prophecies, she reacts by saying that she needs to:
... chastise with the valour of my tongue
All that impedes thee from the golden round,
Which fate and metaphysical aid doth seem
To have thee crown'd withal.
In other words, her mind immediately jumps to a ruthless and unethical conclusion about the best means for achieving kingship. What this suggests is that the couple both had some degree of free will and chose the worst possible means of achieving a fixed goal, and thus that indeed their punishments (insanity, death, and murder) are examples of divine justice.