I would appreciate it if you could let me know whether the following sentence is free of errors.
In actuality, she has applied to some other universities, but she finds your research area so beautiful to forget about! She would very much like to join your research team and I would appreciate it if you let me know of the positions to apply for the once they arise

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It's nicely written but there are a few small corrections I would make:

In actuality, she has applied to some other universities, but she finds your research area [so] TOO beautiful to forget about! She would very much like to join your research team and I would appreciate it if you COULD  let me know of [the] ANY positions to apply for [the] once they arise

Answer

To be honest, she already applied to other universities, but because she finds your research area absolutely fascinating- she would genuinely like to join your research team. I would appreciate it if you could let me know of open positions when they arise.

If you want to sound like an American native speaker, use this above.