Learn Croatian: Imaš li kruha za vrapce? - Do you have bread for the sparrows?

This is the partitive genitive. After verbs like imati (to have) or when talking about food and drink, using the genitive implies an unspecified quantity. Asking Imaš li kruha specifically means 'Do you have some bread' rather than asking if you have a specific, whole loaf.

What is the dictionary form of vrapce, and why does the spelling change so drastically?

The nominative singular is vrabac. When the accusative plural ending -e is added, the fleeting 'a' in the final syllable drops out. This puts the voiced 'b' right next to the voiceless 'c'. To make pronunciation easier, the 'b' shifts to its voiceless equivalent 'p', resulting in vrapce.

Which case is used for vrapce here?

It is in the accusative plural. The preposition za almost always takes the accusative case when it means 'for' a recipient or purpose. For masculine nouns, the accusative plural ending is -e.

Could I say Da li imaš instead of Imaš li?

Yes, Da li imaš kruha za vrapce? is grammatically correct and very common in everyday spoken Croatian. However, in standard and written Croatian, starting the question with the verb followed by the question particle li is usually preferred.