Learn Polish: Czy mamy skuteczną szczepionkę przeciwko rakowi? - Do we have an effective vaccine against cancer?

This is because of the preposition przeciwko (against). It always requires the Dative case. Since the base word rak is a masculine noun, it takes the standard masculine Dative ending -owi.

Could I use the shorter word przeciw instead of przeciwko?

Yes, przeciw and przeciwko mean exactly the same thing and both require the Dative case. You can safely say szczepionka przeciw rakowi, though przeciwko is often preferred in modern everyday Polish because it flows a bit more smoothly.

Why do we use the endings -ą and -ę in skuteczną szczepionkę?

This is a quick Accusative case refresher. The phrase for effective vaccine is the direct object of the verb mamy (we have), so it takes the Accusative case. For feminine singular words, the adjective takes the -ą ending (skuteczną) and the noun takes the -ę ending (szczepionkę).

I've seen the word rak used for an animal. Is it the exact same word?

Yes! The Polish word rak means both the disease cancer and a crustacean like a crayfish or crab. You can remember this by thinking of the English word and zodiac sign Cancer, which is also represented by a crab.