Question : Must declare the table variable "@table_name".

I get the following error:
Msg 1087, Level 15, State 2, Line 16
Must declare the table variable "@table_name".

In the following code:
=============================================
DECLARE c CURSOR FOR
      SELECT table_name FROM vwAgency_tables WHERE aid = @iAid

OPEN c
FETCH next FROM c INTO @table_name
      WHILE @@fetch_status = 0
            BEGIN
                SELECT @iHitCount = COUNT(*) FROM @table_name WHERE TIN LIKE '%'+@sTIN+'%'   --      <---------------
                    FETCH next FROM c INTO @table_name
            END
CLOSE c
DEALLOCATE c
=============================================

Is there any way to do this without using EXEC on a varchar?

The end result is that id like to keep a running total count of matches from several tables (those tables are only known at runtime)

Answer : Must declare the table variable "@table_name".

The code: SELECT table_name FROM ... led me to believe you were looking for tables.

Oops .. I didn't realize the system proc used a cursor. Silly me.

Here's a completely non-cursor alternative though it also uses dynamic sql:

--===========================================================
declare @loop int, @sql nvarchar(4000), @Hitcount int
Create #c table (uid int identity primary key not null, sql nvarchar(4000) )
insert into #c
select 'SELECT @HitCount = @HitCount + COUNT(*) FROM [' + table_name + '] WHERE TIN LIKE ''%' + @sTIN + '%'''
FROM vwAgency_tables WHERE aid = @iAid

set @loop = scope_identity()
set @Hitcount = 0

while @loop > 0 BEGIN
  select @sql = sql from #c where uid = @loop
  exec sp_executeSql @sql, N'@Hitcount int OUTPUT', @Hitcount = @Hitcount OUTPUT
  set @loop = @loop - 1
end -- loop

select @hitcount as Hitcount
--===========================================================