I have a spreadsheet which lists work assignments:

A B
C D
Name Primary Assignment Secondary Assignment Tertiary
Assignment

I have also built a summary sheet which has a box for each assignment
type. I want every person's name who is assigned to a specific box to
automatically pull in when the first sheet (above) is updated.

A B
C D E
Assignment A Assignment B Assignment C Assignment D
Assignement E

Want names to pull in here. Name might appear in multiple columns or
just one.

Any thought on how to accomplish this. I don't think a vlookup would
work in this case.

If a result like this will fill your needs

Assign Aa Ab Ac Ba Bb Bc Ca Cb Cc
Abe 1 __ __ 1 __ 1 __ __ __
Bill __ __ __ __ 1 1 __ 1 __
Chuck __ 1 1 __ 1 __ __ __ __
Don 1 __ __ __ 1 1 __ __ __
Earl __ 1 __ __ __ 1 __ 1 __
Fran __ 1 __ __ __ 1 __ 1 __
Gus __ __ 1 __ __ __ 1 __ 1
Hal __ __ 1 __ __ __ __ 1 1

then you can use Pivot Table.
No VBA code or formulas are required.
Here is the base data:

Emp AsigPri AsigSec AsigTer
Abe Bc Aa Ba
Bill Bc Cb Bb
Chuck Ac Ab Bb
Don Aa Bb Bc
Earl Cb Bc Ab
Fran Bc Ab Cb
Gus Cc Ca Ac
Hal Ac Cc Cb

Use John Walkenbach's Reverse Pivot Table
http://preview.tinyurl.com/2xykg6
to get this from the base data: (partial view)
Emp Assig
Abe AsigPri Bc
Abe AsigSec Aa
Abe AsigTer Ba
Bill AsigPri Bc
Bill AsigSec Cb
Bill AsigTer Bb
Chuck AsigPri Ac

Delete the second column and label the other
columns as shown.
Create the PT shown at the start with
these two columns.
Layout: ROW = Emp, COLUMN= Assig
DATA = Count of Assig
Options: Uncheck Grand Totals