How many shots do I need for statistically valid results?

That depends on the definition of "validity".

The scale in the lower-left corner of the analysis pane gives an indication of the measurements quality. The confidence intervals depend on how actual shots are distributed across actual targets, but the following numbers can be used indicative guideline (give or take a few shots):

To get to minimal acceptable Level 1 quality (confidence interval is 50% of the value, poor quality), 17-19 shots are needed.

Level 2 (40% confidence interval, fair quality): 24-26 shots.

Level 3 (30% confidence interval, good quality): 45-47 shots.

Level 4 (20% confidence interval, excellent quality): 65-67 shots.