It is possible: McCleary v. Washington is an example. Ground zero was the 2012 ruling McCleary v. State, 173 Wn.2d 477, which then took 6 years of further scuffling to resolve. That opinion is full of useful legal tidbits, but the argument boils down to a constitutional obligation (art. IX) for the state to provide an education. If your state has no constitutional provision mandating that the state provide an education, you may be out of luck.
I should point out, though, that the issue reduced to funding and not content / method: that the state used to use a "local pots of gold" model rather that a "big pot of gold" model, and even then came up short of the funds required to do what they were supposed to do. The "argument" was, simply, "We can't can't agree on an affordable means of implementing this system", and subject matter or instructional methods were not debated. A lawsuit will be completely ineffective over a dispute about best methods. Since at least in Washington, school policy is set by an elected set of school officials, the only solution is to pick better individuals next time. Recall is not an option, except in the case of misfeasance or malfeasance – improper acts, not errors of judgment. But I would not totally discount the skills of a clever attorney to make the case that so-and-so is a violation of a constitutional duty, depending on what your constitution demands w.r.t. education.
The constitutional provisions of Washington vs. California are significant. In Washington,
It is the paramount duty of the state to make ample provision for the
education of all children residing within its borders, without
distinction or preference on account of race, color, caste, or sex.
and the Supreme court found that
The State has not complied with its article IX, section 1 duty to make
ample provision for the education of all children in Washington.
Most of the education article is about the funding obligation, which is the legal point on which the Legislature was held in contempt. The California Constitution imposes a weaker duty on the legislature:
A general diffusion of knowledge and intelligence being essential to
the preservation of the rights and liberties of the people, the
Legislature shall encourage by all suitable means the promotion of
intellectual, scientific, moral, and agricultural improvement.
The vast majority of the constitutional provisions in California are about electing and paying administrators, so there is little basis for arguing that a California school district has not encouraged such improvements. There is no "quality" requirement, just a desideratum to encourage improvement.