Just to triple check, and I think the answer is yes, but does your mac (the “host” machine; not the VM) have the bedrock files in example.com/site directory, including the composer.json file? See discussion (and how local_path variable functions). If the files are not there, you need to clone bedrock to that site directory and vagrant reload --provision
If the files (including composer.json) are present in the site directory on your mac (host machine), it indeed suggests a problem with the syncing of site into the VM at current. You mentioned that the “etc/hosts address has been uncommented.” I hope that means you do have a 127.0.0.1 localhost entry in your /etc/hosts, which would affect whether the nfs sync dirs work (details). If you had to uncomment or add this entry, you may then need to repeat some steps you’ve tried such as vagrant destroy, double-check /etc/exports, etc., maybe even uninstall and reinstall vagrant.
As for vagrant log, you could just post the output that appears in your console when you run vagrant up. In rare cases of investigating tricky vagrant issues, you can boost the verbosity of the vagrant output, e.g., with vagrant up --debug (vagrant debugging details), which also mention option to output to log file: vagrant up --debug &> vagrant.log).
I do not see the same composer.json issue when I do a fresh vagrant up with the same specs:
System info:
Ansible 2.4.1.0; Vagrant 2.0.1; Darwin
Trellis at "Fix `raw_vars` functionality for Ansible 2.4.1"