We have a customer with remote sites connected via wireless. I want to setup the ESI-Link between the sites using their existing network. Is this possible?

Site A (ESI100) 10.10.10.201
Site B (ESI50) 10.10.10.202
Site C (ESI50) 10.10.10.203

Gateway 10.10.10.1
Subnet 255.255.255.0

IVC cards from different systems can't be on the same subnet. It might work, but it's not supported because there is layer 1 and 2 taffic between cards on the same system that other systems are not supposed to see. You will need a router in front of each card to route the packets. It should work fine if you have good latency, etc.

I've done it once, I don't remember exactly how but I think the gateway of the local site was the IP address of the second site's IVC card.

That is correct, if you only had 2 systems, but with 3 it is not likely you can do this. I am running into the same problem right now at a customer that has 2 remote sites, and 1 site that connects on the same subnet at the 700 cabinet. For some reason it works with the customers Cisco 1800 router at the 700 location, but they swapped it out for a ASA last week, and took down the 702 (locally connected) cabinet. We are thinking about putting a router in front of the 702 site to see if that will fix their issue.