Eu gostaria de provar a sopa, porque dizem que ela leva alho e cenoura.

Yes. In Portuguese, subject pronouns are often omitted when the verb form already makes the subject clear.

So both of these are natural:

• Eu gostaria de provar a sopa
• Gostaria de provar a sopa

Including eu can add emphasis, contrast, or just make the sentence a little more explicit.

Gostaria is the conditional form of gostar and is commonly used to sound polite or less direct.

• Eu quero provar a sopa = I want to taste the soup
• Eu gostaria de provar a sopa = I would like to taste the soup

In restaurants, shops, and other polite situations, gostaria is very common in Brazilian Portuguese.