falen5,
With what's already built into an SE/30 and a normal Ethernet card that already exists, there's plenty of ways to interface to the modern Internet.
You're simply never going to get a satisfactory web experience out of it. A "modern resource card" is an interesting idea, but most vintage Macs don't have displays large enough to use such a thing. At minimum you'd need an 832x624 monitor, the minimum recommendation for modern computing is 1024x768 or thereabouts (and the LC475 simply can't drive much more than that without dumping down to like sixteen colors) and most people are using, at minimum, 1366x768 or 1280x1024 displays on their modern computers.
And both Windows 10 and mainstream Linux distributions (packaged with GNOME, KDE, etc) are doing a really good job of making 1280x1024 feel cramped.
Your best bet is to consider telnet on the local LAN for text-only interfaces to things. Basically, do it the way you actually would have done it in the late '80s or early '90s and dial directly to a UNIX system and use that to get on the Internet.
This isn't necessarily strictly impossible, but it's realistically beyond the collective abilities of everybody here. Even Cameron Kaiser wasn't able to modernize existing code for Mac OS 9 that much, and that's without the limitations of RAM, CPU power, and without saying "oh, well, let's just put an ARM CPU on a NuBus card and offload this application's work to that card" -- which is clever, but at that point why not just run the simplest X11 browser you can find and run that on a Pi or a PC and display it on the Mac?
And that still doesn't address the issue of so many web sites absolutely demanding large displays these days.
At the end of it what you've got is something like a Quadra 650 or 800 (or IIfx/950) with several ARM coprocessor boards doing particular tasks or sharing the workload of different browser instances, with a big LCD hooked up to the biggest baddest GPU you can find, which you own't really be using for its Mac capabilities as much as for it's ability to drive a modernly large monitor, and at that point why not just buy a Griffin iMate for your ADB kb/mouse and connect it to a modern PC minitower with Linux, using MATE as the graphical interface?