Do, the initial drop test is the Copper Sulphate test. Its a solution of Copper Sulphate, specific gravity 1.05something. A drop of blood with a Haemoglobin greater than 12.5g/dl will sink. So by failing the initial drop test, this would appear to indicate your Hb is lower than 12.5.
The reason they try to find a vein if someone has failed the drop test, is to follow the usual procedure of taking a venous blood sample for at the least a Haematocrit measurement, more probably a Full Blood Count, which measures several clinically significant parameters. This, in turn, should establish whether anaemia is actually present, and if so ,what the likely type / cause is.
Taking a mulitvitamin with iron might possibly help if for example you were actually anaemic, and the reason for the anaemia a shortage of iron - but even more important in that case is to find out why you are short if iron. Dietary insufficiency, familial trait in not processing iron, bleeding ulcer / chronic bleeding syndrome. Each cause may require a different treatment response.
Being able to access the vein has little to do with any hypothetical anaemia. Thats just down to your venous physiology and the expertise of the staff.