As others have stated, it can be done and it's important to not overlook the technicalities or accidentally overstep the boundaries, finding yourself brokering transactions when you didn't intend to. Most contracts have an assignment clause, so you may want to work with a good real estate attorney initially and have them review documents, provide representation, or possibly prepare a statement or addendum for you to use on any wholesale deal that you are participating in. Practicing attorneys can assist clients purchase or sell property for a fee in NM.
The practice of wholesaling is not new here, but also not widely known about or understood, and is an idea or approach to investing that many dabble in, but few stick with it. Be proactive about connecting with experienced professionals, whether legal, real estate or otherwise; and stay in your lane. That can be easier said when there aren't a lot of clearly defined rules. Over the years, I've been contacted a few times by homeowners who had been in contact with wholesalers, who had definitely overstepped boundaries and were offering services that require a real estate license or other professional services. I don't think it was due to bad intentions on the part of the wholesaler, but that they found themselves working with a willing homeowner with some hurdles to clear. Without intending to, the wholesaler was in the position of having to help the homeowner find housing to move to or was handing out legal advice in instances where the homeowner had a financial hardship or when the property was part of an estate with beneficiaries that had differing opinions. Obtaining your real estate license in NM, if you planned to mostly invest here, can help expand the boundaries that you can legally work within, but does not eliminate them or the related risks altogether.
1) In order to activate your real estate license, you'll be required to purchase an Errors & Omission Policy, which is insurance coverage for agents and brokers and is intended to help pay most legal fees and claims made against you by a third party for damages and malpractices. Most E&O policies do not provide coverage for an agent's personal dealings outside of their personal property that they occupy. Sometimes additional coverages can be purchased, and I'm not sure if or how this might impact a licensed agent who was wholesaling their own deals. Claims made against agents, related to personal dealings, have an average award or settlement amount that is 5 times greater than claims made against an agent, related the property of a 3rd party. That is why there are limitations or exclusions on coverages there, and also why there is a greater financial risk for continuing to handle them yourself when you don't actually have E&O coverage.
2) You will also be required to "hang" the license with an actively licensed qualifying broker and brokerage in NM, who essentially oversees your real estate transactions and is responsible for making sure that you are practicing real estate within the state & federal laws & regulations. Brokerages have policies on handling personal transactions and dealings, so you'll want to explore options there and be upfront with how you intend to use your license. After a couple of years and certain amount of experience, combined with additional training, testing and fees, you can become a qualifying broker yourself, and have your own brokerage, but initially you are required to work at a brokerage, for another broker. There are continuing education requirements and fees to maintain and renew your real estate license, as well as fees to join the local Realtor associations, MLS, etc.
If you are mostly interested in becoming an investor, and using wholesaling as your way of getting there, I'd start by connecting with a good real estate attorney first, and then there is lots that can be learned online as far as best practices.