When were tantalum capacitors first used in computing?
There's a fair bit of history and references in the Wikipedia article, so it's worth exploring those. 

We were certainly using tantalum capacitors in industrial electronics long before the personal computer was available, and I'm sure it was used in military electronics as well. 

Very early (pre-IBM, eg. S-100) desktop computers probably had a few tantalum caps in them, there appears to be one in this photo of a static RAM board (the blue input cap for the local linear voltage regulator), but most of the bypass caps appear to be ceramic. They would have been sold in the mid-to-late 1970s.