A woman tries to engage with her dysfunctional biological family after she learns that her late loving and caring mother abducted her as a baby.

Hmm... Well it seems you have identified a compelling theme with, "is blood thicker than water?" being the dramatic question of your story. As far as the logline is concerned I would double down on pitching that core aspect of your narrative because that is where the heart of the story is. In which case it will make more sense to describe the protagonist as a cancer patient (as opposed to "a woman"). By deliberately getting the "illness" aspect of the story mentioned in as few words, you are essentially allowing more room to explain the family dynamic as it relates to the theme of the story.

Before going any further in developing the logline I just need a few details regarding the plot clarified:

1) Is the loving mother alive at the start of the screenplay or has she already passed away by page 1?

2) Does the mother appear predominantly in flashbacks, or is she referenced more-or-less in conversation?

3) What is the inciting Incident?

4) Does the ending answer the theme of "is blood thicker than water," or does the ending leave this question open-ended and/or ambiguous?

Here is the story, so far. A woman is raised by a single mum that had a really great job. Paid for a private school, a really nice middle class life. The woman has never felt anything but loved. She knows she is adopted, but this has never worried her.

Years after her mother dies, she needs bone marrow. She goes on the register and also gives DNA to find out if she has any relatives. This is when she discovers she was abducted by her mum from this other family.

This family are highly dysfunctional. They don’t support each other and are the type of people she would cross the road to avoid. She goes and meets with them. The venom and hatred they have for her mother is huge and justifiable. But still hard for her to come to grips with. She knows they have been wronged, but she also can see what her life would have been like.

This tension makes it hard for them to bound. They feel betrayed by the love she has for her late mother.

That is the story. Is blood thinker that water? Is she willing to betray the memory of the woman that loved her and raised her for a chance that she will have a donor?

I think the, "hoping to get assistance with her medical issues," bit sounds a little bit vauge and it makes the protagonist's actions come off as a bit passive. (She "hopes" being the main culprit of making her goal sound passive.)

If anything you might want to swap this detail and maybe focus on the dynamic that the protagonist's "mother" was loving whereas the biological family are not.

If not then you might have to omit that detail too in order to best explain the medical situation in the logline in a more meaningful manner. Example:

"A young woman must convince her rejecting biological family to sign her up for an expensive experimental medical procedure that she's too young to legally sign herself."

I tried to explain both the dynamic between the "late loving abducting mother" and the "dysfunctional biological family" but this dynamic, when paired with the nature of the medical issue bloats the logline to nearly 40 words. It seems, in order to keep that word count concise, that focus must be made on either the "abduction dynamic" or the "medical issue". Only then can a sense of focus be implied to the logline.

With the revelation that her late mother abducted her, a woman struggles to connect with her dysfunctional biological family hoping to get assistance with her medical issues.

I like the premise but I feel like the logline might benefit by having the inciting incident (or so I assume) occur at the start of the logline.

Here's an alternate logline for you consideration:

"With the revelation that her late mother abducted her, a woman must confront an even harsher reality as relations with her biological family rapidly deteriorate."