How can a blood test give information about rickets?
From the Wikipedia article I linked to in your question:


  Serum calcium may show low levels of calcium, serum phosphorus may be low, and serum alkaline phosphatase may be high from bones or changes in the shape or structure of the bones.


There is also a rare genetic form of rickets called Vitamin D-resistant rickets or X-linked hypophosphatemia that can be diagnosed by DNA sequencing of cells collected by a blood draw.

It should be noted that, at least in the case of rickets, blood tests alone will not diagnose the disease. There are many other possible symptoms of the disease, so a patient presenting with one or more of these symptoms may also be subjected to a blood test to confirm the diagnosis, not just make it on its own.