Q:

Adobe Illustrator file to STL (STereoLithography) file

I have made a vector graphic in Illustrator and would like to know if it is possible to add a depth to it and convert it into an STL file suitable for 3D printing purposes?

A:

Not in Illustrator alone, but any 3D program worth its salt can do that. What you want to do is:

Export your graphic from Illustrator as .svg
Import the .svg graphic into your 3D program
Extrude your graphic to desired height
Export that as .stl

Personally, I use Rhino3D, but it has a bit of a steep learning curve and is pretty pricey. Use whatever .stl capable 3D software you are familiar with. If you never used any, try something like the free and newbie-friendly tinkercad. I managed to make a quick extrusion of my logo in less than a minute. 

When you upload the .svg to tinkercad, it will automatically add depth to your 2d graphic. You can adjust the depth by dragging different handles. 
Few things to keep in mind:

For best results, make your graphic into a single shape (I am not sure how tinkercad handles multiple shapes in one extrusion, but if they are not a single solid, you might run into problems when printing).
Bear the size in mind. When importing, tinkercad will ask you how big you want your shape (in millimeters). This is the size the object will be printed. Depending on where your print, you might be able to change the scale just before printing, but better safe than sorry. 
Be careful of tiny details. Depending on the printer used, there will be minimal size the detail can print. If you go below that size, the printer will just create a blob over there. 
as user287001 mentioned, when 3d printing, you will lose everything but shape (so color, raster textures, etc). 

