You can include Ambient Occlusion in an exported image. However when exporting an image from a view with Ambient Occlusion enabled there is a threshold related to the pixel size of the export and whether or not the effect can be included. Typically once you go above 2000-2100 pixels in the horizontal direction the effect will be excluded.
For example; you have a monitor resolution of 1600 X 1200. You are exporting a shaded view with Ambient Occlusion enabled, at 300 DPI, to a TIFF image. If you keep the R > Export > Images > Images and Animations > Image > Image Size, around 2000 pixels, Ambient Occlusion should be included.
Go above +/- 2050 pixels [which can vary based on your monitor resolution] the effect should be excluded from the image export. I have included 2 comparison images below: