I guess those "Internet sticks" connect to the USB port... The problem with using USB may be because the older eMac models had USB 1.1 (not 2.0). If so, 1.1 is probably too slow to support an Internet-connection device, even if the software is able to run on the eMac. I'm not sure which model you have, but if you can ID it from this list, the profile page will provide the specs (including USB port).
Can Vodafone (or an alternate service provider) provide you with a more typical "modem" device, that has an Ethernet port on it? Then, you can probably use your eMac's Ethernet port to connect to it using a standard network connection (no special software needed).
First of all, would it be correct to assume that everything regarding installation and connection is exactly according to the mobile broadband modem instructions for your very operating system (see below for a link to the K3770 device guide page)?
Vodaphone should be able to inform you about any necessary software/firmware updates (these may be carried out automatically).
Generally speaking, if a computer for some reason cannot connect to the Internet because of system software version limitations (which does not appear to be true here, since Leopard seems to be supported) for a mobile broadband stick, a workaround will become necessary. One way is to use a dedicated mobile broadband router (with support for the USB mobile broadband modem and the network in question) with Wi-Fi capabilities, and then connect from the computer to the router via Wi-Fi. Another way could be to use a MiFi device, which combines the mobile broadband modem and the Wi-Fi base station in one small unit.
EDIT: I just noticed Kenichi Watanabe's comment about the USB port speeds. This could of course be a limiting factor as well. Another one could be that the USB port is unable to provide enough power to the device. In both cases, a mobile broadband router or a MiFi unit may be of interest instead.
>Vodaphone should ...
Vodafone should ...
Below just one example to show what a mobile broadband router with Wi-Fi capabilities is.
A MiFi device is sometimes known as a Wi-Fi hotspot. One example below.