Looks like that you may not be using an updated version of that UDF, which you can find here:
The author did an update specifically to support what you're looking for, though they only provided an example with the Enter and ESC keys.
I was interested in this, so I did play around with it and managed to add tabbing functionality, though it's not great. It also really only works if you set "All" on both the Columns and Rows. It may be possible to extend it further to loop through and look for the next available editable field, but that's more work than I'm willing to try out on this.
Give this version that I updated a shot, and see if it works. Again, make sure to set your ListView to 'All': __ListViewEditInput_AddListview($Form1, $List1, "All", "All", "E")
One annoyance with it was that for me, when I tabbed to the next column, it didn't visually 'highlight' or select the text in the cell. However when you tab into a cell it does have the WHOLE cell selected, and whatever you type will replace what's in the cell by default. Again, you might be able to change it around some to resolve that.
ListViewEditInput.au3
Edit: Actually I'm dumb, I didn't check the actual example that @Kanashius provided in that linked post. In that example he did add a function for supporting tabbing himself, I haven't tried it out, but it looks a lot simpler than my approach. Either way, check out the original authors updated file and example, or my updated one. Both should have what you're looking for. Mine also drops to the first column on the next row when you tab while at the end of a row.