Does Islamic law permit a retired, ill person to reclaim the costs of their treatment and medication from the National Social Security Fund, given that membership in it was compulsory during their government employment?

There is no sin upon the participant in compulsory social security; rather, the sin is upon the one who compels him. The participant must rid himself of the forbidden portion of what he receives. There is no objection to your relative recovering medication and treatment expenses from the fund, provided these expenses are less than or equal to his capital.