I suppose it means that everyone has a lawyer, or can get one quite easily if the need arises.

"Lawyer up" is a reflection of the "litigious" American society today. It describes any situation that persons involved in a conflict prefer to "hide behind their lawyers and legal technicalities" instead of making progress toward resolution of the problem. So it goes beyond an accused criminal having access to a lawyer. It also means that parties to a commercial conflict do not try to resolve the disagreement, and instead "dig in their heels" by letting lawyers cite all sorts of complicated legal justifications, each holding on to its side of the argument and end up not getting anything done. In American English, this term can be used in formal writing, as did the writers of the New York Times.

One small suggestion that differs from Brian's answer. If you are quoting the New York Times, it would be fine to use the phrase "lawyered-up", but it remains a slang/informal way to state that one has ready access to lawyers. The New York Times is not "immune" from slang (or from quoting slang). If you are writing your "own" academic paper, and you are not quoting someone else, do not use it.

Thank you very much, Amy! I got it!

Hi Hana, "Lawyered-up" is a slang/informal way to state that you have ready access to a lawyer. You would not use it in formal English.