Sorry, the redirection into variable doesn't work as expected. The following, inserting it straight from shell, works fine.
Let me format it nicer:

Code:

sh$ logFile=someLog.txt
sh$ awk '{
  print "Searching for "$0 > "'$logFile'"; 
  printf("grep -il \"%s\" * | sed \"s/^/%s, /\"\n", $0,$0); 
}' Customers.lst

Input:

Code:

sh$ cat Customers.lst
John Doe
Alice In Wonderland
John Other

Outputs:

Code:

sh$ logFile=somOther.log; awk '
{print "Searching for " $0 > "'$logFile'" ; printf("grep -il \"%s\" * | sed \"s/^/%s, /\"\n", $0,$0); 
}' Customers.lst
grep -il "John Doe" * | sed "s/^/John Doe, /"
grep -il "Alice In Wonderland" * | sed "s/^/Alice In Wonderland, /"
grep -il "John Other" * | sed "s/^/John Other, /"

So awk is used to print out a bunch of commands with your names from Customers.lst. Nicely each command on one line.
Now what does each command do when you execute it:

Code:

grep -il "John Doe" *

will search for "John Doe", case insensitive (-i), in all files present (*) and output the filename only if string is found (-l).
So if you have 3 log files present in current directory:

Code:

sh$ ls
file1.log
file2.log
file3.log

and only one -- file2.log -- contains "John Doe", then:

Code:

sh$ grep -il "John Doe" *
file2.log

that's the filename that grep returns.
Now you want <name>, <filenameThatContains_name>
so that's what sed does; it inserts 'John Doe, ' in the beginning of line ('^'), so:

Code:

sh$ echo file2.log | sed "s/^/John Doe, /"
John Doe, file2.log

Those '%s' are format specifiers for printf, it's telling printf to print string, and $0,$0 are arguments to printf, telling it to substitute $0 (which is the whole record -- the name (if you don't have anything else on line of Customers.lst) in awk) for %s.

So the awk magic is just to format and print out a nice command to the screen. Then you can test out one of those commands, if you wish, or just assess them visually (useful for debugging, before you actually run them).
Then pipe to bash to execute, and redirect to capture output.
You might need to insert a directory name for grep's argument, instead of just plain '*' to adjust for details of your dir structure.

I know it's not quite elegant, but should be faster.
Loops in shell are not nearly as efficient. Awk excels in the speed of reading lines from input -- it's a filter, after all, well crafted for this particular purpose. Then you are gonna launch a process for each grep; this can be optimized further (parallelized onto the CPUs, e.g).

If you want, you can do a little benchmark, take a subset of your logfiles, and process them with our script and mine.
Run it with 'time' like:

Code:

time ./scriptToRun.sh

which will spit out the time it takes for script to finish. I'd be curious...
Also I can't wait for answers of more experienced *nix people.