Nerve conduction study
and Neurologist's opinion
This is an unfortunate accident, rare although.
Yes you should see a Doctor.
The needle to draw a blood are usually no 23 and thin enough not to cause a major injury
to a nerve. Yet it is a better idea to show to a neurologist and get the nerve conduction study ASAP to know the extent of the problems and to start on an appropriate therapy.
I hope this will solve very soon, the nerve conduction study will guide further to know the recovery.
Please do not try anything at home.