What are the consequences of hiring someone who allegedly signed a non-compete form?

This is a very dificult problem to deal with.  The consequences are that the company in SC files an injunction action against your company in NM court.  The goal of the action is to prevent the empoyee from working for you.  You seem to believe that there is no issue here based on the nature of the work the new employee is going to be doing.  However, i can assure you that the company sees you as a threat.  You can employ this person, but be advised that you could face some potential claim - whether the claim has merit or not will depend on the noncompete.  If there is litigation, you will be able to obtain a copy of the NC and determine what if any merit the company in SC has. 

It's very odd that the employee does not have a copy of the non-competition form, and that her former employer will not give her one.  Most states require people who make contracts with each other -- and a non-competition agreement is a contract -- to deal in good faith, and this sounds very questionable to me.

I am not a New Mexico or South Carolina attorney; one place you can look for qualified lawyers is our website, http://attorneypages.com  It might not be a bad idea for your prospective employee to talk to an attorney in this field.

You -- and your company -- did not sign the non-competition form, and it does not sound like you compete with the former employer anyway.  However, you might consider whether losing the ability to do business with the former employer would create problems for you, if you do not have a contract that prevents them from simply dropping your company as a client.  As long as you make no attempt to get confidential information of the former employer, from your prospective employee, I doubt that you have anything to worry about, although it is always best to talk to your own lawyer, with more of the details, to be sure.

In most states, non-competition agreements have to be reasonable, and limited in both how long they are in force and how much ground they cover both in terms of geography and possible future jobs for the employee, because everyone has a right to make a living.  The nature of the employee's work does have a bearing on what is "reasonable."