Short answer. Possibly, but only one office at a time. Each local civil registrar keeps a registry book of the applications filed with him, in the order received, showing the applicants' names and the date a license was issued. The Code creates a local book, not a national index.
What the law says
The local civil registrar concerned shall enter all applications for marriage licenses filed with him in a registry book strictly in the order in which the same are received. He shall record in said book the names of the applicants, the date on which the marriage license was issued, and such other data as may be necessary.
Family Code, Article 25 — Registry Book of Marriage License Applications. Read the full provision →
What the registry book actually holds
Article 25 requires each registrar to enter all applications for marriage licenses filed with him in a registry book strictly in the order in which the same are received, recording the names of the applicants, the date on which the marriage license was issued, and such other data as may be necessary. Two features of that design matter to your question. The book is chronological rather than alphabetical, and it is the book of one office. It answers 'what was filed here, and when', which is a narrower question than 'has this person applied anywhere'.
Applications are not marriages
Keep the two records apart in your mind. An application recorded in the registry book shows that someone asked for a license; it does not show that a marriage followed. A person may apply and never marry, and a licence may lapse unused. Conversely, a marriage celebrated in another city leaves its trace in that city's records, not in the book where the application was filed. So a clear book in one municipality tells you very little on its own, and an entry in it is not by itself proof that your partner is married to anyone.
The practical route, and its limits
Because the record is local, a search needs a locality: the city or municipality where either party habitually resides, which is where Article 9 says the application must be filed. If you have reason to think a particular place is involved, that is the office to approach, and it is worth asking in advance what it requires of a person requesting a search, since access to civil registry records is governed by rules outside the Family Code. Without a candidate locality there is nothing in this article to search, and no provision of it creates a nationwide lookup.
If what you actually suspect is an existing marriage
That is a different and more serious question than an old application, and it is the one worth resolving before a wedding rather than after. A marriage contracted while a prior marriage subsists is void under Article 35, and the consequences reach property, children and the criminal law. The document people usually want here is a certification from the national civil registry as to whether a marriage record exists for a named person. If that certification comes back showing a marriage your partner has not told you about, stop and take advice before anything further is filed.
Related provisions
- Family Code, Article 25 — Registry Book of Marriage License Applications
- Family Code, Article 9 — Where a Marriage License Is Obtained
- Family Code, Article 35 — Void Marriages