I can't duplicate this. Yes I have an old VB6 setup but we went with MySQL and here we are 16 years later and it still works.
You went with Access so I can't help you directly but what about the ACE? Noted at http://www.dreamincode.net/forums/topic/320397-vb-6-not-recognise-access-2010-accdb/ (you'll have to find the ACE for your version of Access.
PS. http://answers.microsoft.com/en-us/office/forum/office_2013_release-access/help-with-office-2013-professional-and-vb6/59654760-6091-40db-b296-395e2d0cccda?auth=1 notes the 32 64 question.