Learn German: Können wir am Feiertag ein Treffen vereinbaren? - Can we arrange a meeting on the public holiday?

The word am is a contraction of an and dem. In German, when you talk about things happening on specific days, dates, or parts of the day (like der Feiertag), you use the preposition an. Because it answers the question "when?" (a fixed point in time), it requires the dative case. This changes the masculine der to dem, and an dem shrinks down to am.

I know treffen as a verb meaning "to meet". Why is it capitalized and used as a noun here?

In German, you can easily turn an infinitive verb into a noun just by capitalizing it! Here, the verb treffen (to meet) becomes the noun das Treffen (the meeting). These "made-from-a-verb" nouns are always neuter, which is why it uses ein in this sentence.

Why is the verb vereinbaren all the way at the end of the sentence?

This happens because of the modal verb können (can). When you use a modal verb, it acts as the "boss" and takes the main verb position. Since this is a yes/no question, Können goes at the very beginning. The other verb, vereinbaren (to arrange), is pushed to the very end of the sentence in its original infinitive form.

Could I just say Können wir uns am Feiertag treffen? instead?