I was given an iPhone 1 back in mid-2008 when a friend bought the iPhone 2. I'd like to send the phone back to Apple for a battery replacement. There is no SIM card in the phone, as I was using it as an iPod and to surf the web over WiFi only.

Will Apple accept this iPhone 1 from me if I pay for the battery replacement service or will they refuse?

More than likely, they'll accept it.

Why not simply give them a call and find out for sure?

Well if you are in North America then look at Lithium-Polymer iPhone Replacement Battery
for all Apple iPhone "1G" (4GB, 8GB, & 16GB models) (http://eshop.macsales.com/item/NewerTech/BIPH1G1400M/) might be something you would like to try.