Scabby, dry bald patches on my week-old puppy's back
That's about all there is really. The patches amount to no bigger than a centimeter in total, no vomiting or diahrrhea observed, area does not smell.
Could I put some pure aloe vera on this?
Before you put anything on these patches I'd strongly recommend having your vet take a look at your puppy.
There are all sorts of possible reasons for puppy hair loss including parasitic issues such as Mange, fungal infections such as Ringworm, or allergies, hereditary conditions and more. Only your vet can make an accurate diagnosis and recommend treatment.
Best of luck, hope you get this taken care of and your pup is doing better soon.