Learn Polish: Ci wolontariusze muszą sprawdzić te bilety. - These volunteers must check these tickets.

In the plural, Polish splits nouns into two groups: masculine personal (groups including at least one man) and everything else. Since wolontariusze (volunteers) refers to people, we use the masculine personal pronoun ci. Bilety (tickets) are inanimate objects, so they fall into the "everything else" category, which uses te.

Are both Ci wolontariusze and te bilety in the Nominative case here?

No, they are in different cases! Ci wolontariusze is the subject performing the action, so it is in the Nominative case. Te bilety is the direct object receiving the action, so it is in the Accusative case. For non-masculine-personal words like bilety, the Accusative plural looks exactly the same as the Nominative plural, so it stays te.

I thought masculine nouns usually end in -y or -i in the plural. Why is it wolontariusze?

That is a great rule of thumb, but masculine nouns that end in "soft" or "historically soft" consonants (like sz, cz, rz) often take an -e in the Nominative plural. Since the singular is wolontariusz, the plural becomes wolontariusze.

Why use sprawdzić instead of sprawdzać after muszą?