Dear Bob,

The first concern I have is what that pest might be?  Most pesticide labels tell you exactly what pests they will work for, and will tell you how long you have to wait before eating the plant    , or in this case the fruit. But the key factor to know is what pest it is.  

Here is a link to some grape pests:

Grape Insects | Entomology (uky.edu)

Here is another:

10 Worst Grape Vine Pests: How to Identify and Treat Them - Minneopa Orchards (minnetonkaorchards.com)

From what I can tell, cutworms may eat young shoots of grapes.  The second article has some information on dealing with cutworms.

It is very difficult to determine exactly what treatment is needed without knowing the culprit.  It is possible that deer may be eating off the shoots as well.  I hope you can identify the pest to be better able to determine the management that best suits the problem. Additionally I have a factsheet which should provide some general information on growing grapes in the home garden.

Growing Grapes in the Home Fruit Planting | Ohioline (osu.edu)

Please feel free to contact me again if you have additional questions.  

Sincerely,