Is English grammar important to learn English?

Vocabulary is the most important. Grammar is just the standard operating procedures for using vocabulary. Without vocabulary, you wouldn't even understand what grammar tells you.

I guess the most important thing is to be able to communicate. The catch is that to communicate you need to know grammar. As soon as there is a subject and a verb in a sentence there's grammar. If you're thirsty, you can always say "Water!", but as soon as you try to make something more elaborate (Can I have some water?) you need grammar. So it's not a question of whether grammar is important or not because you have to learn it anyway.