Can user create a record If he don't have access on object and executing apex class(Creates new record on that object) having with sharing keyword?

Can someone help me for this below scenario?
1. Track a Primary Contact on the Account object.
2. Track a Primary Contact on the Opportunity object
      a. This contact should default to the Primary Contact on the Account.
      b. This contact can also be over written manually by user.

If user don't want to use salesforce Id's and want to load junction object records by using Import Data Wizard with Parent record names only. 

Can user create a record If he don't have access on object and executing apex class(Creates new record on that object) having with sharing keyword?