Learn Farsi: آنها دارند مداوم می‌دوند.

This is exactly how adverbs of manner work in Farsi! You can often take a plain adjective like مداوم (continuous) and use it directly to modify a verb without changing its form. Farsi doesn't always need an equivalent to the English "-ly" ending; its position right before the main verb shows it's acting as an adverb.

Why does the adverb مداوم go right in the middle of the verbs دارند and می‌دوند?

In the present continuous tense, the auxiliary verb دارند (they have) and the main verb می‌دوند (they run) act like a frame. Adverbs of manner usually sit directly before the main verb they describe, which naturally drops مداوم right between the helper verb and the main verb.

Could I use به طور مداوم instead of just مداوم in this sentence?

Yes, you absolutely could! به طور مداوم literally means "in a continuous manner" and is completely correct. However, using the bare adjective مداوم by itself is shorter, very natural, and extremely common in both spoken and written Farsi.

Just to refresh my memory, why is دارند (they have) used when the English translation says "they are"?

Farsi uses the conjugated forms of داشتن (to have) as helper verbs to create the present continuous tense (actions happening right at this very moment). So, دارند doesn't mean "possession" here; when paired with می‌دوند, it creates the specific meaning of "they are currently running."