It is hard to challenge the Texas law because it was written in a novel way: It effectively deputizes private citizens to enforce it, and bans the state government from doing so.
The law empowers private citizens to file lawsuits against both abortion providers and anyone who “aids and abets” abortions, which could include clinic staff members or even people who drive women to clinics. If such a lawsuit succeeds, the plaintiff can win a $10,000 judgment plus legal fees, and the judge can impose an injunction barring the defendant from performing or aiding any additional abortions. If the lawsuit fails, the plaintiff does not have to pay the legal costs of the defendant.
This structure means there is no obvious and specific person who can be the defendant in a case challenging the law on its face. Typically, a legislature that wants to restrict abortion in some way writes a law that the state would enforce, such as by prosecuting doctors or rescinding their medical licenses. Abortion rights supporters can then challenge such laws by suing the officials who would be responsible for enforcing them, seeking court injunctions ordering those defendants not to enforce those laws.
With the Texas law, however, there is no obvious defendant or target of an injunction. In the parallel cases before the Supreme Court, a group of abortion providers and the Justice Department have tried instead to sue Texas judges, clerks and other state officials. The central question for the Supreme Court on Monday is the technical legal issue of the legitimacy of that approach.