How do medical labs find vitamin deficiency in blood tests?

To determine the amount of vitamin in the blood. If it's too low, than it's probably a deficiency

It depends which vitamin you are referring to as different vitamins have different tests for them most of the vitamin deficiencies are diagnosed based on clinical features  .... but i will tell you Vitamin B12 which is regularly done to diagnose megaloblastic anemia

Serum VitB12 assay : Two methods for measuring vit b12 concentration are available a microbiological and radioisotope assay . The principle of the microbiological assay is that the serum to be assay is added to a source of vit b12 to a medium containing all other essential growth factors for a b1 dependent microorganism eg : euglena gracilis
The principle of the radioisotope assay is the isotope dilution of the non radioactive vit b12 by added Co-57 labelled Vit b12 . A carrier binding capacity is then used to adsorb a portion mixture of radioactive and non radioactive vit b12 . The free and bound forms of the vitamins are separated and the quantity of radioactive vit b12 adsorbed to the binding substance is measured . By comparing the measurements of a series of standards of known b12 is measured