Is it permissible to cancel a sports club membership and get a refund after the company has covered the subscription fee, with the intention of gaining financial benefit from this service?

The fundamental principle in these matters is to refer to and abide by the conditions set by the granting entity, based on Allah's saying: "O you who have believed, fulfill [all] contracts," and the Prophet's (PBUH) saying: "Muslims are bound by their conditions." Therefore, refer back to your company and ask them; if they grant you permission, then there is no harm.