Will Contacts Sync Between Procore & MYOB?

Yes, however due to the way that MYOB processes their contacts, they will need to be either a Customer or a Supplier.
If a company is both a customer and a supplier, you will need to make a customer version of this contact and a supplier version of the contact and then map them separately.

Also, the integration will only automatically sync contacts that are a Company, not an Individual. So, when creating a contact in MYOB they will need to have this designation: