Suppose that your brother confided in you that he had embezzled $20,000 dollars from his employer to help pay for his son’s medical bills. If he had not stolen the money he would not have been able to afford to both pay for his son’s care and his mortgage payment, and so he would have been forced to choose between not providing medical care for his son or having his family lose their home. (He has a wife and a daughter as well.) He tells you that you are the only person who knows about this crime and you think it extremely unlikely that he will be caught unless you turn him in. He works for very large company, which has not noticed the missing money yet, and in the unlikely event they did discover the missing funds, he has covered his tracks quite well. What would virtue theory say you should do in this situation? Would a virtuous person report the crime to the police? Why or why not?

Virtue ethics has to do with a focus on achieving eudaimonia or human flourishing through development of one's ingrained moral character. Thus one concerned with virtue ethics asks of decisions whether they are in accord with this moral character and whether they contribute to developing or undermining it. 
The first thing to note about the statement of the problem is that it includes many pieces of information that philosophers concerned with virtue ethics would consider fundamentally irrelevant. Whether your brother is likely to be caught or not would be of interest to a consequentialist but irrelevant in the case of virtue ethics. Like a deontologist, a virtue ethicist would be concerned with the quality of the actions themselves rather than whether one is or is not likely to "get caught."
The main issue virtue ethics would raise is the effect of these acts upon character. If your brother has been engaging in practices including theft and deception, misleading his colleagues and feeling that he is somehow entitled to steal because he thinks that his personal needs outweigh the needs of all other people, he is acting in a way that harms his moral character, leading him to develop the habits of dishonesty and stealing. Getting away with such acts reinforces the bad moral character he is developing and passing along to his children. 
In terms of your own position, complicity with lying and stealing is itself wrong, but so is lack of love and compassion for one's family. To act in accordance with virtue, you might try to persuade your brother to turn himself in and offer to loan him the $20,000 to repay his company when he does so. This would help improve his moral character and help you develop the moral character of generosity, empathy, and honesty. 
https://plato.stanford.edu/entries/ethics-virtue/