- This とこ is a colloquial variant of ところ, which is a very flexible noun that means not only "place" but also "time", "situation", "aspect" and so on. Haven't you seen these patterns where ところ refers to something related to time or moment?
- Yes, 入れて is the te-form of 入れる ("to put in"), but why "progressive"? The te-form does not express the progressive aspect on its own.
- This やる is not する, but an informal and less polite variant of あげる. やる and あげる are subsidiary verbs, i.e., it has a special function after a te-form. As you may already know, (-て)やる and (-て)あげる are notoriously difficult to translate directly, but here it's basically just "for them (the new members)".
- You're correct that なきゃ is contracted なければ. A super-literal translation of 名前を入れてやらなきゃ is "if I don't give (a favor of) putting a name". But なければ, なくては, なきゃ and so on can effectively mean "have to".
- This って is not a topic particle but a quotative particle used without an explicit verb. Here something like 思って is implied. See: verb+ようにと、 or verb+かと、
Here's a translation (added punctuations for clarity):
新入部員も入ったことだし、「名前を入れてやらなきゃなー」って(思って)、今書いてんだ。
Now that the new club members joined, I thought "I gotta put their names (for the sake of them)", so I'm writing them now.