Everyone is under surveillance all the time these days. There would have to be some very secure way for women to contact any organization trying to help without raising any flags that could be used to tip off the police. This would be a real challenge.

Of course a modern underground rail road would function different than the one 300 years ago.

Certainly, but I suspect there will be parallels. I consider it important to understand the past and apply that knowledge to the present to predict the future.

Legal adults don't need to notify anyone that they are moving.

Of course not. But you know as well as I do how important personal rights (particularly women's) are to the party of Law and Order. If women started fleeing, they'd find some line of arguing why this couldn't possibly be them exercising their human rights.

Multiple normal states have already passed laws that protect women from any persecution for laws restricting their bodily freedoms.

My worry is that the other states (or rather the unhinged citizens thereof that create such legislation that would warrant saving women from them) may escalate this ideological conflict. We've seen the hateful incitement of violence happen already. It's not legal to storm Congress either.

If I don't respect your laws, believe that they're "wrong" anyways and disobeying them is right and just, and think that I can get away with it, what's to stop me?

no way to tell why a woman left.

Obviously evil librul indoctrination! The same people believing that the female body "has a way of shutting that down" and that a woman's proper place is in the kitchen probably don't respect the woman's actual motives, if they consider it an evil plot. As you rightly pointed out:

Because their goals are to subject women and to ensure that there's always a poor, unhealthy, uneducated, and subservient population.

If they don't believe on a woman's right to self-determination, none of the good and legal reasons women might want to determine their own lives will matter to them.

It's way easier to move a willing person over someone who is surrounded by a supportive community. I'd imagine this organization would help LGBTQ+ people escape as well.

And that is exactly what the opposition would need to be: a stronghold, united in the purpose of being a safe haven for all the oppressed.

All these parallels to slavery aren't an accident. Like the Railroad mentioned in the premise, I worry that an effort to liberate women from the stranglehold of a society trying to subjugate them, yet depending on them, would spark a similar counter-effort to recapture them. In the hypothetical that such a modern Railroad would be created, this Railroad and all its supporters would similarly need to be prepared for a violent response. I wish freedom could triumph peacefully, but history suggests poor precedents for that.

That's not an attempt to dissuade, mind you. I'm all for it. Liberate the enslaved and oppressed from their oppressors and have a plan to defend them.