I cannot say 100 percent but coolers are a very important part of life for Panamanians, kind of like a car in L,A, so my answer would be yes. If you could not bring a cooler than no Panamanians would take the ferry.

Yes, I'm pretty sure that you can take a cooler with you.

Update, took Sea Las Perlas ferry from Amador, Panama City to Contadora Island. Yes, we saw some coolers on the boat. Sea was a somewhat rough, keeping my eyes closed and taking a nap helped sea sickness. I did not throw up. Originally sat outside then moved inside. So glad I did as the waves lapped against the boat and people who sat outside looked like they took a shower by the time we arrived to the island.

People brought coolers on the Ferry when I visited the Island 2 weeks ago. I will definitely be bringing a cooler on my next Contadora trip. Keep in mind there are not too many buying options when you're on the Island so if a cold cerveza on the beach is a priority then bring a cooler. You can buy beer and wine and ice but not a cooler.

They did not seem overly strict or concerned, like weighing your baggage when boarding the ferry, but it's up to you to transfer your belongings from the ferry to the small shuttle, and from the small shuttle to the beach, so bringing too much is not recommended.

What is the cost of the ferry round trip and does it run multiple times each day or just once there and back?

The are two companies who do the ferries. Their fares and schedules are on their site:

https://ferrypearlislands.com/en/online-booking/

https://booking.ferrylasperlas.com/en/home/find/#/reservar