The application I'm currently working on requires three ports to be opened. At the moment these are 5024 through 5026 but on reading around I discovered that these lie in the ICANN registered range (i.e. ports < 49151).
Is there any need for me to inform any organisation that I plan to use these ports if it's within a local network only or can I just go ahead and use them?
You can just go ahead and use them. The only way you'd need to notify ICANN is if you were going to have a significant number of clients using those ports in general use; in that case, you would want to try to register the application with the port to prevent potential collisions with other applications using those ports.