Hi, Sherry: I'm giving you the address for LDS, just click "catalog" and then do a "place" search using term "Tallapoosa":
You'll see that there are many court records but nothing which specifically indentifies "Minutes of Court, Applications for Letters of Administration" covering your period of interest so you'll have to do some sleuthing. I would go to the microfilm "General Index to Records of Tallapoosa County" and hopefully that will reference a specific film. My best guess is that the document you seek might be in the 87 reel collection: "Tallapoosa County Estate Records 1840-1915" but that is only a guess. There is also an entry for "Tallapoosa County Probate Court Minutes 1850-1851; 1857-1862, etc." which indicates a gap from 1852-1856. However, the film(s) "Orphans Court Books 1850-1852" states that it includes probate court minutes, I presume for those years. There are other possibilities but I'll leave it at that. Again, I would start with the general index for all records.
You made me curious so I checked out the Alabama Department of Archives and History website and in their catalog under "Local Government Records Microfilm Database" it sure suggests that they have this material. Their database gives a description of reel content, box location, reel #, whether it can be duplicated and an order number if one wants to purchase a copy ($40). They do state that most of the originals are with the county. IF you live in Alabama you might want to visit your local library and see what help they can offer. Sometimes libraries can obtain microfilm on loan so that might be worth considering. Also, have you called the courthouse in Tallapoosa to see what they say?
Don't know if any of this helps but would like to know how this quest shakes out.......Rick