고자 is a connective ending used when the speaker has a certain purpose, intention, hope, etc. 고자 하다 means "intend to do (something)" 고자 하는 is a modifier form of 고자 하다 e.g. 그 모임은 잘못된 제도와 관행을 개혁하고자 한다 The group intends to reform a wrong system and custom. 그들은 잘못된 제도와 관행을 개혁하고자 하는 모임을 만들었다. They made a group which intends to reform a wrong system and custom.

-고자 is an adverb form, so it is always used with a verb. -고자 하는 is an adjective phrase modifying a noun. In your example, you could also say just 하고자 without 하는. 1. 잘못된 제도와 관행을 개혁하고자 모임을 만들었다 = ... created a group TO REFORM the bad system and practices (개혁하고자 works directly with 만들었다). 2. 잘못된 제도와 관행을 개혁하고자 하는 모임을 만들었다 = ... created a group (which is) AIMED AT REFORMING [which has the purpose of reforming] the bad system and practices (개혁하고자 하는 modifies the noun 모임) (The difference is like that between 사람이 많이 왔다 (N + adv + V) and 많은 사람이 왔다 (adj+N + V)) The two are more or less the same in broad meaning, but #2 puts more focus on the group itself. -고자 is a formal phrase which is used in the -하고자 한다 / 하고자 하는 form a lot (typically in formal writings and official speeches). In real life, we use the simple -려고 (한다) or -기 위해(서) most of the time.