I am on a Corporate networked computer and it won't allow queries or macros. I need to count values in one column only if they match with values in another column. Is there no formula (i.e. countif) that can do this for me?

Hard to understand the question. Are you talking about value in A1 must match A2 to be counted, then B1 must match B2 to be counted, etc? Always a separate instance of matches?
A1 = A5
B1 = B5
C1 = C5

Or, are you allowing multiple matchings in one column to the same cell in another column?
A1 = A5
B1 = A5
C1 = A5

Lastly, how deep does this list go? How many matches are you possibly counting here? What are you doing with the matched data? Are you applying a new formula to each instance of matches to do more work or are you just looking for a total count of matches?