You can select a slicer then hit TAB twice so there is a dotted square around the first value of the slicer, then you can play around with arrows, CTRL, SHIFT, SPACE. If you keep hitting tab you can also browse to other slicers.
I am adding to this old thread because I arrived here when trying to figure out how to activate a slicer without using the Mouse. I finally figured it out. It is very subtle and I wanted to document it somewhere so I would not forget it.
In my example, I have a pivot table in A3:B17. I added a Sales Rep slicer, then an oval shape, then a star shape, then a Product slicer. I am starting from a cell in the pivot table.
1. Press Alt, H, FD, and then the letter O to enter Select Objects mode. Not much appears to happen, but the cell pointer changes to a white arrow instead of a black cross.
2. Press Tab once to select the Sales Rep slicer. At this point, you've selected the container for the slicer, so the arrow keys will move the entire slicer .
3. Press Tab again to move into the slicer, first to the Multi-Select icon added in Excel 2016. Spacebar will toggle the Multi-Select state.
4. Press Tab again to move to the Clear Filter icon. Press spacebar to clear the filter.
5. Press Tab again to move into the slicer items. This is where I was trying to get. Be very cautious here because you don't automatically move to the first slicer tile. You move to the last slicer tile that was touched. How can you tell which slicer is active? There is a light dotted border on that dark blue slicer and it is almost impossible to see. Press the arrow keys a few times and maybe you will see the subtle formatting moving. Now that you are in the slicer tiles, using arrows will move among them. If you enabled Multi-Select, then pressing spacebar will toggle the selected slicer item. If you forgot to enable Multi-Select, then choosing one slicer tile will turn off all the other tiles. From here, you can Shift+Tab once or twice to move back to the Clear Filter and Multi-Select icons.
6. Press Tab again to leave this slicer. In my case, I move to the Oval shape.
7. Press Tab again to move to the Star shape.
8. Press Tab again to move to the Product Slicer. You are now essentially at step 2 again.
When you are done selecting from slicers, you have to get out of the Select Objects mode. Press Escape and you will return to the last selected cell in the pivot table.