Your attempt has two grammatical errors:
- "To bite on the neck" is 首を噛む.
- There's no need to use the ta-form for 噛まれる (because of the "relative tense" rule).
After fixing these, we have:
彼女は彼が首を吸血鬼に噛まれるようにした。
This is okay, but for better readability, you may want to change the word order:
彼女は彼が吸血鬼に首を噛まれるようにした。
Note that this sentence means she had no direct control over 彼 and 吸血鬼. She only "set up the situation" (by shutting a door, etc). And it does not state whether or not the vampire actually bit him. In other words, she only attempted. If you need to clearly express that the vampire actually bit him, you'll have to rephrase the sentence, for example:
彼女は彼を操り、吸血鬼に彼の首を噛ませた。
彼女が○○したことで、吸血鬼は彼の首を噛んだ。