I was wondering, which is the correct sentence in this case.
你的生日是几月几号？ or 你的生日几月几号？ (without 是)
Also can I omit 的 in this case?
The line without 是 is not a correct sentence because it doesn't have a verb. It's fine in spoken language but should be avoided in formal writing.
In this sentence, it is fine to omit 的 in spoken language, too.
你的生日是几月几号？ Grammatical, fine for both formal and informal usages.
你的生日几月几号？Not a complete sentence, only used in spoken language.
你生日是几月几号？Whether it's grammatical is arguable. Regardless, it's common in spoken language.