Whether marriage is legal or illegal is off the topic. However, a religious marriage can never be illegal. What I think you are trying to say is that it is NOT recognized in the secular world. So if you ONLY have a religious marriage and do not get a marriage certificate you are not recognized as a married couple for any legal reasons such as insurance, next of kin, medical decisions, etc.
Rav Moshe z”tl was very adamant about getting a marriage certificate and he himself took care of mailing it in. He was our mesader kiddushin, and he would not shtel a chupah without it.