I have a question about marriage: Is it biblical for a man and woman to get married simply by taking a personal vow between them and God, or must it be done legally?
What creates a marriage is a covenant. "Yet she is your companion and your wife by covenant" (Malachi 2:14). A covenant is a special type of vow which is made before God. It has numerous parts (see "What are covenants?" for details) and isn't just used for marriages, though marriages are the most common type of covenant used.
In a covenant, there are several parts:
A definition of the parties involved and their status. In a standard wedding ceremony, you will see the preacher ask each person if they desire to marry the other. He will also ask if any knows a reason why the couple should not marry.
A history of the parties' relationship. Often here the preacher discusses the reason for marriage.
The obligations expected of the weaker party to the stronger party. Often the stipulations involved promises of loyalty to the stronger party. In a marriage ceremony, it is a union of equals, but there are still stipulations. The "for richer or poorer, in sickness and in health," etc. layout the stipulation that each will be there for the other no matter what the circumstances.
An agreement where a written copy of the agreement would be retained and the provisions for periodic reminders of the covenant. This is where the state license comes in. They serve as a repository for the fact that a marriage covenant was made. In the older days, it was often kept at individual churches, but this was often awkward if the evidence needed to be located because the record could be in so many different places. Over the years, state governments in the United States took over the duty. Periodic reminders are done by our anniversary celebrations.
Who or what would be able to confirm that the covenant was made. Often these are composed of people who witness the covenant. In a standard wedding that is the function of the best man and the bride's maid, but it is also why we typically have a wedding ceremony. All the guests are there to be witnesses that the marriage took place. Under covenant laws, a physical witness was also used since people do die. In the United States, we typically use rings to act as a witness that the person is married.
Curses and Blessings:
The consequences of breaking or keeping the covenant. In a typical wedding, more emphasis is placed on the blessing of marriage, but you will find somewhere in the ceremony the statement "So help you God" which alludes to God sitting in judgment over the keeping of the covenant.
The sealing of the covenant. Theses are the vows made during the ceremony and are often combined with the stipulations and the exchanging of the rings.
After a covenant is completed all parties are invited to sit down at a common meal. The meal is to represent the fact that all differences have been resolved and that the parties are at peace with each other. In a wedding, this is done by the reception.
Personal vows are just not on the same level as a covenant. Many people want to avoid the recording of their marriage by the government. They imagine all sorts of nonsense regarding giving up rights, but the reality is that the government is serving as a repository of the covenant. Nothing more is done with the information. And since Christians are to obey the laws of the land when they are in accordance with God's laws (Romans 13:1-2), there is no reason no to meet the civil requirements for a marriage.