Here's a link to child size standards: http://www.yarnstandards.com/childsize.html.
If you know the gauge and the measurements for the size you're making, you can determine the number of stitches for major measurements (such as the chest). For example, for a child size 8 with a 26.5" chest and a gauge of 5 sts per inch, you'd need 132.5 sts (26.5 x 5) around the chest to fit exactly. However, since it's a vest you'd want 1-2 inches of ease. So, the chest would be somewhere between 27.5-28.5" or 138-143 sts. Find how many stitches the pattern says to cast on and set up the following proportion:
(#chest sts size 8)/(#chest sts in pattern) = x/(#CO sts in pattern)
Cross mulitiply and solve for x. Then you'll know how many sts to cast on.
There are many things to consider, though. If you start out doing 2x2 ribbing or a pattern that requires an even or odd number of stitches, that influences the number of stitches you cast on. If you start out with 2x2 ribbing in the round, you'll need a multiple of 4 stitches to cast on.
Some of it is strictly math using proportions, some of it is winging it! At least for me it is. I recently re-wrote an adult pattern for a toddler starting with the chest measurement and going from there. I used proportions and the yarn standards size chart for children. Hopefully, you'll get a lot of suggestions from many other knitters here!