Make sure you have an entry for each letter.
Do a find for A
Make a script for Find A
Do a find for B
Make a script for Find B
This is tedious, work on a backup of your database, save AS often, test often
In FMP 10, you could use one script for all the buttons and pass the letter as a parameter, choosing a different letter parameter for each button.
In FMP 5, you don't have that option.
You could make one script and simply duplicate it, giving each script a different script name. You could name them "Search A", "Search B", "Search C" .....
Write your script like this:
Enter Find Mode
#I may not be remembering the exact name of this get function in FMP 5, but there is one that returns the name of the script
Set Field [YourSearchField; RightWords( Get ( CurrentScriptName ) ; 1 ) & "*" ]
Set Error capture [on]
Perform Find 
# do what you want to do with the found set here
Once you have the script working for finding all records where name starts with A, simply duplicate this script and change its name so that the last word of the script name is the appropriate letter of the alphabet.
Note: This script is formatted as best I can remember for FMP 5, FMP 7 and later releases would need to include a table occurrence name in the Set Field step.
Thanks for you help.
I had tried this and was doing something wrong. Following your steps, carefully, I was able to make it work.