Everyone who's been trying to renew a CHL knows it's hard to get an appointment. Mine expires in July and I've been trying for months to get an appointment at Lake county. I live in Cuyahoga, but the Lake office is way better to deal with. But the appointment scheduler has always returned "no availability".
Yesterday I had an inspiration and waited until midnight. At 11:59 pm I hit the "schedule appointment" button, no dice. But as soon as the computer clock struck 12:00 midnight I hit "schedule appointment" again. Bingo, the schedule for June 30th appeared. I booked an appointment and received a confirmation email. I don't know how long they lasted but as of this morning there was no availability again.
So if you're looking for an appointment in a county with an online scheduler, try waiting until exactly midnight.
Edit - corrected date from June 31 to June 30