Learn Polish: Chciałabym zarezerwować ten duży pokój.

The ending -abym tells us that the speaker is female. If a man were speaking, he would say chciałbym. Both are polite, conditional ways to say 'I would like' and are much softer than saying 'I want' (chcę).

What form is the verb zarezerwować in, and why?

It is in the infinitive (the basic dictionary form meaning 'to book'). Just like in English, when you use a phrase like chciałabym, the action verb that follows it stays in the infinitive.

Since the room is the object being booked, shouldn't ten duży pokój change its endings for the accusative case?

It actually is in the accusative case! The verb zarezerwować requires the accusative, but because pokój (room) is a masculine inanimate noun, its accusative form looks exactly the same as its basic dictionary form. Because the noun doesn't change, the describing words ten (this) and duży (big) don't change either.

Why use zarezerwować instead of the basic verb rezerwować?

Zarezerwować is the perfective form of the verb, which means it focuses on a completed result. When you book a room, you want the action to be successfully finished, so we use this form instead of the ongoing or repeated rezerwować.