When the page posts back, the search value will be in the text box. Construct the new query as you describe and get the value from <textboxid>.Text. You'll probably want to wrap the search string in % so it will find the search string within the column value instead of expecting it to be exact.
You'll probably want to build this query using a SQL command object so you don't have to worry about the SQL injection risk. You can add a parameter into your query, then add the search value to the command's parameters collection. Something like this:
sqlCommand.CommandText = "SELECT * FROM dbo.Bids WHERE Active='1' AND DATEDIFF(day,OpeningDate,GetDate()) < 1 AND BidName LIKE @BidSearch ORDER BY BidCategory, BidName, ExpirationDate"
sqlCommand.Parameters.Add(New SqlParameter("@BidSearch", String.Format("%{0}%", <textboxid>.Text)))
-Peter
peterlanoie.blog