Internet access has nothing to do with connecting to a wireless access point. Even if your internet connection is cut off entirely, you'll still be able to access the local area network.
If you're getting an error message to the effect that it's timing out while obtaining an IP address, the problem is probably with DHCP. In that case, you set yourself up a static IP address on the PSP so it doesn't need to bother obtaining one.
1: When you come up to "Address Settings" in network setup, select custom, and at "IP Address Settings", select manual.
2: I suggest bringing up a command prompt in Windows (Start, Run, and type "cmd" and press enter). At the command prompt, type "ipconfig" and press enter.
3: ipconfig should give you most of the necessary information. Give your PSP a slightly different IP address. I have my PC set to a static IP address of 192.168.1.67, so I gave my PSP 192.168.1.68 because I knew it wasn't taken by any other device. Basically you only want to change the last number, the first three should be the same as on your PC.
4: Enter the next two values exactly as they are in the corresponding slots (Default Gateway is the same as Default Router) on the PSP.
5: For primary and secondary DNS, you can either find your ISP's DNS server addresses (probably in your router settings somewhere) or, if you're too lazy, use these two: 208.67.222.222 and 208.67.220.220. These are maintained by
OpenDNS and are free for anyone to use.
Once you're done with that, the rest of the settings should be easy. If you don't know what a proxy server is, you probably don't have one, blah blah...