My condolences on the loss of your friend.
The term “legal document” really doesn't have much meaning here. What matters is whether the document your friend left is a valid will (or part of a will). In IL, among other requirements, a valid will must be signed by the person making the will (called the testator) and the signature must be witnessed by two people over age 18 who also sign the will attesting that they witnessed the testator sign the will. It is apparently not necessary, however, to have the will notarized. These requirements are similar to those used in a number of other states as well.
A will that the testator writes entirely by hand and signed by the testator is called a holographic will. Some states provide a special rule for holographic wills that recognizes them as valid wills even though the will was not witnessed as described above. However, it appears that IL is not one of these states. Instead, it appears that IL requires that a holographic will be executed in the same manner as any other will (i.e. that it be signed by two witnesses).
The signature of the notary may count as one witness. But if that is the only witness to the will, it would appear that the document you have may fail to qualify as a will under IL probate law. If that is the case, you likely have no claim to the horse, unfortunately.
I suggest that you take the document to an IL probate attorney to find out for sure whether you might be able to make a claim for the horse. There may be other details here that could change the outcome, and an IL attorney can tell you with better certainty where you stand. Since I've not seen the document and do not practice in IL, I cannot tell you for sure that you don't have a claim to the horse.