Webhosting should not affect your Authorize.Net integration especially if you're using DPM. DPM sends users to Authorize.Net's servers for payment and then back to you transparently so your host doesn't really need to be able to do anything special. If a webhost supports PHP then they're good enough to support DPM and Authorize.Net.
I use Godaddy a lot, if you have no idea where to start. Their tech support isn't -quite- as helpful as it was some years ago, but the stuff that comes with an account works fine and you probably won't need tech support unless you're doing something really complicated.
I'll add my two cents and say I've used Hostgator for dedicated servers, reseller hosting, and shared hosting and never had any issues and their technical support is excellent.