Without language would culture exist? Possibly, but to have culture, you must have some way of communicating with others. The deaf use sign language, but it is still a form of language. The possibility of developing a culture without language of some kind is slim at best. By language, I mean communication with one another. It is truly a prerequisite of cultural development between people.
A second reason I believe language came first has to do with my religious beliefs. God created Adam and Eve, they were the only one's in existence for a time, and probably developed a 'culture' of sorts, but they were able to speak to one another from the beginning. Therefore, language came first, and then culture developed as they had children after they were expelled from the Garden of Eden. As more and more people populated the earth, and especially after the tower of Babel when language was mixed up, various cultures developed within groups that remained together, separating them from their 'cousins' in a sense.
Those who were then separated by the language barrier, drifted into different places with those who spoke a like language. They developed their own cultures and became distinct from the others because of the commonality of their language. When the language changed, the culture changed. That tells us that language, common language, drives the creation of cultures. Whether the language is oral or written or otherwise, it is a necessary ingredient to the development of culture.
On the other hand, that communication does not have to be oral; it just must be understood between the different peoples of a culture. The deaf culture is a great example of how a people who cannot speak can develop a culture all their own. Even though the deaf are individuals just like the rest of us, there are certain things many of them have in common. They have developed a "deaf culture" that is unique from that of the hearing world. This is true across languages and across countries. Their common lack of oral language translates into a culture seldom understood by the hearing.
Another example is the culture that develops within families, especially with twins or close siblings. They develop a culture of sorts that is unique from all others and yet possibly similar to others in the same situation. The culture develops from a language, but it is not a language recognizable to outsiders. It involves looks and gestures and what may look like mind reading at times. Language, whether oral or not, drives communication which drives the creation of shared culture. Our ability to communicate with one another allows us to create cultural bonds with others. It also allows us to expand our culture to share the culture of others. Language makes the world go around.