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.
How big of a project is it? If there are more than a few thousand internet-exposed users, you may want to investigate informing them somehow. If its on a closed LAN of any size, it doesn't matter at all.
the language in the relevant RFC is that applications SHOULD NOT use unregistered ports. Nobody's going to sue you if you use them; the only issue you'll get is that if someone else decides to (1) use the same port (2) not register it, you'll confuse the heck out of each other. As McWafflestix says, you can go ahead and use them.