When you are viewing the PiAware SkyAware page do you have a value for Position History and ADS-B Message Rate? If other than zero is it changing? Make sure the boxes for ADS-B, MLAT, Other and TIS-B are selected.
Does the top right of your map page look like my screenshot minus the red arrow? If it does click on the box that my arrow is pointing to (Toggle Sidebar).
This next screenshot will show the boxes below the aircraft data that will select what data source you want to view on the map. If that does not work post a picture of your map page and maybe we will see something else that would be a clue.
The web server should be running already as it is part of the install. It sounds as if you might have a local network problem of some sort.
Where are you trying to view the local map from?
Is it from a computer on the local network?
Can you ping the PI4 from that computer?
Also is the pi connected via a WiFi or a wired connection?
Is the computer you are trying to access it from WiFi or wired?
If one or both are WiFi do you have isolation enabled in your router?
Something like this:
It was a standard install from the SD card. It worked perfectly except for the local web server on the PI4, so when wiedehopf gave me the tip above for tar1090 now it works, no probs at all and very fast.