Okay I thought I read this one way and then read the replies and read it another way.
You have 3 tasks
1) update the ESXs hosts
2) update Application server - presumption here is that they are clustered already
3) update SQL server ... its not clustered
For
1) shared storage move your VM's from the Host you are going to patch and move them back once finished
2) If this is already clustered .. it some sort of load balance app server that can have 1 application server down. Then its not a VMWare problem, just shut down one server let the load balancing look after it. VMWare can help you it can snapshot for easy roll back if you want. If its not clustered... then you could take a clone of the machine and run it whilst you up date the original ... but you can't have them both on the network at the same time .. This again is not really an VMWare question. The application developers should be able to provide help here
3) If its static data, then clone it and work on the original but same problem you can't have both on the network at the same time. If its not static data ... you could use a shared lun and 2 copies of SQL, but that's fraught with danger, better to have a SQL cluster or SQL replication. You are going to have to accept some down time on your SQL server, only patch when needed. Or investigate SQL replication ... you will need to SQL server or 3 depending on the SQL replication option you use
You could for SQL do 2 VM's use a VIP for the SQL ip ... that way you can move the IP and have both machines online at the same time. you will need a shared LUN .. thats the tricky bit ! and have SQL installed on both machines. so patch one and then make it master.. so remove the SQL data lun from the other machine and associate it with the patched machine, then restart SQL and off you go ... there is still down time ... and you have to manage a lun that is shared between 2 servers.
I hope you not charging them too much for this ?