Should I ask a professor for the third time for a letter of recommendation?
I was in a similar position as you a few months ago. 

Given the deadline, it matters that something rather than nothing is submitted before the deadline. Ask asking for extensions is an unreliable strategy. 

Send an email to your professor stipulating that you are concerned about the lack of response (which you are) and inform the professor what your plans are (asking someone else). 

No need to ask why, be short and to the point. 


  Dear Professor X,
  
  It has been sometime since the last time we talked about you writing a
  letter of recommendation on my behalf. I am concerned that your
  silence is indicative of your unavailability to write a letter. I know
  you are busy and I understand your time constraints. 
  
  In the meantime I’ll be asking [you can choose to be specific or
  vague]. 
  
  Thank you for your time and consideration.
  
  Best regards,
  
  [Your Name]


Make sure the second letter writer is willing and able to submit in time.