After reading Alice's comments, I started making some phone calls to see if it was possible for me to deal with the Social Security Number issue personally.
The hospital in which M was born confirmed that they only file for an SSN if the parent fills out a certain piece of paperwork while there. (This part of the information took several days as it was hard to find the right number to call for the answer and several people told me to talk to someone, only to have that someone never answer their phone and not respond to voicemail messages. Probably because that someone wasn't the right person to be asking in the first place!) This procedure seems to vary from state to state, as in my state (and in TX, apparently) the filing is automatic. (In my state, in fact, there's a space on the birth certificate for the SSN.) In M's Birth State, it appears to vary from hospital to hospital, as I found other hospitals in that state whose websites say that they will do it "automatically." M's hospital had no information on the website, but when I finally got to talk to someone who knew anything, she told me that BioMom would have had to sign something specific for them to file for it.
No way to know for sure, of course, whether BioMom did that or not, but I'm willing to guess that she didn't. That means we're looking at applying for an SSN, not just requesting a replacement card for an SSN that already exists.
I called the Social Security office that serves M's county in my state. They told me that I could not file for the SSN at this point because I am not M's legal guardian, the state is. So, although they would be allowed to issue me a replacement card with the paperwork Alice mentioned (my placement papers, M's birth certificate and shot record, etc), I cannot request a number for her until the adoption is final.
I thought about calling the Social Security office in M's birth state, but, frankly, it's over 2 hours away and I'd have to get state permission to take M over there. Also, I suspected I would get the same answer.
On the bright side, I'm told that the adoption caseworker has a colleague going to the Social Security office late this week and they will be filing for a number for M. So, hopefully, that will be wrapped up soon.