Most longer zoom camera have a shutter lag of about 0.5 seconds.
Which is probably a bit too long for your purpose.
And there are no cameras with 8X zoom and image stabilization.
I did find one camera with a shutter lag of 0.2 second and it has an optical zoom of 12X. It does have image stabilization.
Here is a link to a full review of the Kodak Z612:
There are sample photos on the last page of the review.
My wife and I want a new digital camera. The one we have is not fast enough for our two grandchildren. We want a camera that runs on AA rechargeable batteries, has an SD card, and an optical zoom of at least 8X with image stability. The smaller the better. Is there such a camera out there with an almost immediate picture capture? These two kids move way too fast!