I'm sorry about this situation. I suggest that you find someone else who is willing to write you a letter within a short timespan.
Talk to the programs to which you have applied to understand how they will consider a late letter of recommendation. Many programs are very lax as to when a letter of recommendation arrives, but will begin sorting and analyzing candidates the day after the deadline (some even before!). Tell the programs, either by phone or by e-mail, that your reference has just recently informed you that something personal has occurred and that it is unlikely they will be able to provide you with a letter, but that you have already arranged with another person to submit it. Ask about how and if your late recommendation will be considered, and whether your application will suffer from it. Also ask how you can include an additional reference in the system if it is not directly available through the system.
I know that some programs will look into candidates even if their file is incomplete. However, they will only officially admit you once all documents are in place. Some will reach to you to let you know that you are missing a document, but the best way to know is to ask them directly.
If you don't ask someone else to submit a letter, the worst case scenario is that your application won't be considered at all. If someone else submits a letter, the worst case scenario is that you may be competing with a lukewarm letter, which is still better than not competing at all. If the original reference submits their letter, you will have four letters, which is better than two strong and one lukewarm, but not much worse than three strong, and significantly better than just the two.