Q: There is someone whose testimony would benefit my horse case, but they would be uncomfortable testifying in court and they don’t want to get involved. Can I just get them to sign a notarized affidavit and use that in court?

If a witness has testimony that is necessary to your case, you can make them come to court and testify by having them served with a subpoena. However, if the witness lives far away from the court house, they might be out of the subpoena range.
Another alternative is to do a telephone or live deposition of the witness with a certified court reporter taking down the testimony. This tends to be expensive, but the deposition transcript and excerpts from the deposition are generally admissible in court even if the witness does not show up. If you do decide to do a deposition, you must give the other party notice so that they can ask questions at the deposition as well.