We are a daily and we put out a quarterly magazine that goes to the circulation plus its own free distribution.  Would a writer be eligible to enter column writing if they wrote in all four issues?  The category for daily papers states the column must appear “weekly” but in this case, it doesn’t apply.

No, she’s not writing a column often enough to qualify. If she writes a column anywhere else in the paper, that might get her enough publications to make it fly.