I don't know the story, but I expect that it is this meaning:
- To make a solemn declaration ... to the honesty or truth of ... a declaration.
Many verbs can be followed by a that-clause: you can check in the Cambridge Dictionary to see whether it is permitted for a particular verb. In such cases, the word that should always be written, but it is often omitted in spoken english. The passage is in quotes, so it's supposed to be spoken english, and a that has been omitted. It might be easier to understand if you add the [that]:
What if pure-bloods and half-bloods swear [that] a Muggle-born's part of their family?