What is the ruling on purchasing from a company affiliated with a company named "Trinity Group," and does using its products constitute disbelief (kufr)?

If what is purchased from it and dealt with is permissible, then there is no harm in that. The name of the parent company and the beliefs of its owners are not relevant, because dealing with a non-Muslim is a permissible worldly transaction, not forbidden in principle, such as selling to him, buying from him, lending to him, and giving gifts, for the Prophet, peace and blessings be upon him, used to engage in these matters with disbelievers.