If you're staying at the HoJo's across from DLR you will get free Internet access there - as long as you have an ethernet card on your laptop. That's the only place nearby I've known to have access.
<font color=navy>I don't know of any Internet access at either of the parks or Downtown Disney. I've stayed at both the Grand Californian and DL hotels and had Internet access in my room, but that's about it.
As for LAX, I've only been in two terminals - 1 for Southwest & 6 for Delta, and didn't see any Internet access there.
Actually, I was trying to find a cyber cafe type place - so I don't have to lug my laptop. Most hotels now have high speed internet access (all you really have to have is a phone plug for dial up!).
Thanks for your help! Happy New Year!
<font color=navy>I don't know of any internet cafes nearby, and when I stayed at the GC & DL Hotel I plugged my laptop into their phones, so at that time, it was dial-up (this was about a year ago).