Learn Polish: Czy mogę prosić o państwa dowód osobisty?

The word państwa is the formal plural equivalent of "your". It is actually the genitive form of the collective noun państwo (ladies and gentlemen / Mr. and Mrs.), but it acts as an unchangeable possessive pronoun here. You use it when speaking politely to two or more adults, whereas you would use wasz for friends or children.

Why is the preposition o used here?

The verb prosić (to ask, to request) pairs with the preposition o when you are asking for a specific thing. The phrase prosić o always forces the noun that follows it into the accusative case.

Since prosić o requires the accusative case, why does dowód osobisty look like it is in the basic dictionary form?

Because dowód (ID/proof) is an inanimate masculine noun. For inanimate masculine nouns, the accusative case is identical to the nominative (dictionary) case. So dowód osobisty does not change its ending here.

Since państwa addresses multiple people, why is dowód osobisty (ID card) in the singular?

Just like in English, you can ask a couple or group for their ID in the singular if you only need one document to process a shared reservation, for example. If the speaker specifically needed a separate ID from every single person, they would use the plural: państwa dowody osobiste.