Learn Croatian: Vrlo brzo se šminka. - She puts on makeup very quickly.

It is a verb! While šminka is the Croatian noun for makeup, here it is the third-person singular present tense of the reflexive verb šminkati se (to put makeup on oneself).

Why is the reflexive pronoun se placed after Vrlo brzo instead of just after Vrlo?

Enclitics (unstressed words like se) generally go in the second position of a sentence. In Croatian, the second position can be after the first individual word, but it is often much more natural to place it after the first complete phrase. Here, the adverbial phrase Vrlo brzo (very quickly) is treated as a single block in the first position, so se comes right after it.

Could this sentence also mean The makeup is very fast?

No. If šminka were the subject noun, the sentence would need the verb to be (je), and the adjective for fast (brza) to match the feminine noun, like Šminka je vrlo brza. The presence of se is your biggest clue that we are dealing with an action (a reflexive verb) rather than a noun.

How would I change this sentence to say she puts makeup on someone else?

You would drop the reflexive pronoun se because she is no longer doing the action to herself. Then, you simply add the person receiving the makeup as a direct object in the accusative case. For example: Vrlo brzo šminka prijateljicu (She puts makeup on her friend very quickly).