The free plug-ins for Majestic and Moz no longer exist. What can I use now to make my forum backlink project work?

Unfortunately, you will have to subscribe to any of their plans to make use of the add-ons. OR
Here are the 3 best sources for backlinks:
#1. PBN's 
#2. Web 2.0's
#2a join http://facebook.com/groups/kallzubacklinkexchange
#3. Order Majestic backlink report for your larges NATIONAL competition and steal backlinks.

Please note that the idea is to keep moving. You can come back to the forums but know they are NOT required.