Learn Farsi: آنها دزد را پیدا می‌کنند و می‌گیرندش.

It is attached right to the end of the second verb! The verb is می‌گیرند (they catch). The -ش at the very end is an object enclitic meaning "him", "her", or "it". Instead of using a separate word, Farsi often glues these short pronouns directly to the verb.

How exactly is می‌گیرندش pronounced with that ending?

It is pronounced "migirandash" (or informally "migirandesh"). The base verb می‌گیرند ends in the consonant sound "d", so we use a short "a" (or "e" in spoken Farsi) to connect it to the -ش sound smoothly.

We learned that را marks the specific object. Why isn't there a را for "him" after the second verb?

The word را is only used after independent nouns or full pronouns to show they are the specific direct object, just like we see with دزد را (the thief). When you use an attached pronoun ending like -ش, it already acts as the object on its own, so no را is needed.

Why is "they find" two words (پیدا می‌کنند) but "they catch" is just one verb word (می‌گیرند)?

پیدا کردن (to find) is a compound verb, which literally means "to make visible" or "to make found". Most verbs in Farsi are compounds like this. On the other hand, گرفتن (to catch or to take) is a traditional simple verb, so its conjugated form می‌گیرند is just one word.