I think by "web interface", the tech support person was answering the question (which is what I think you are asking), can I interface the H2-DM1E over the Internet. Based on your Internet access at the PLC, the answer is definitely YES.
If by "does it have a web page for configuration", the answer is NO. The Do-more is a PLC, not a "simple" device like an EBC, ECOM, or ERM. In order to configure a Do-more, you must use Do-more Designer, the PLC programming software. But, as mentioned in the previous paragraph, can work across the Internet. There are a few issues dealing with latency and timeouts when going across THE Internet (vs. a local office network).
See this FAQ on the various ways to make our Ethernet products accessible over the Internet (it discusses the latency/timeout issues). Although this FAQ is for accessing DL PLCs via the ECOM100 using DirectSOFT, the steps are VERY similar when using Do-more Designer to a H2-DM1E CPU. If you do have questions (because the steps are not identical), definitely post them here.
http://www.hosteng.com/FAQFiles/ECOM.htm#FAQ0016