Once you've set up Turabian citation style and configured any given document to use it, Zotero should do the right thing automatically, provided the database entries do have access dates.
Folks the Chicago Manual of Style may not strictly *require* the date of access for web sites and for on-line journal articles but it allows for it. Different disciplines and Universities require slightly different variations of the style. Examples of how to present the date of access are on the CMS Website: http://www.chicagomanualofstyle.org/tools_citationguide.html. There should be a variation of CMS with this included. I assume that there is already some way of changing date formats to suite the region.
Upgrade Storage
Once you've set up Turabian citation style and configured any given document to use it, Zotero should do the right thing automatically, provided the database entries do have access dates.