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.
As a general rule, if you use a 是, you should have a corresponding 的.
这是很好吃的！ 我今天是很忙的。 你今天是怎么样的呢？
The Chinese, though, favor saying less to get the same meaning across. So when speaking, sometimes the 是 and/or the 的 will not be spoken.