This may be the wrong forum, but I can’t seem to find the correct place to ask. I noticed on the MSPMentor website that Avast CEO mentioned a new MSP program for Avast for Business. How does an IT consultant like myself sign up for the MSP program, and move all my current clients over to Avast for Business through the program?
Which article are you referring to? If you mean the article http://mspmentor.net/managed-security-services/avast-launches-free-cloud-security-solution-msps from February (not exactly new), then Avast for Business is the MSP program 
You can sign up via http://www.avast.com/business