I have not seen any web based solutions. As you would have to pipe the audio to and form the radio to the internet site, I would expect you would need software for the audio 'bridging'. If you have a old computer, the data requirements are minimal-many use their 'old' computer after buying a new one.
Another option is to set up a Raspberry PI -although the money to make one 'complete' does get you close to the price of a low end computer.
Another option would be to check out some of the 'surplus' sites (like tigerdirect.com) if you go that route just make sure that you buy everything you need for a system (monitor, mouse, keyboard, ect).
I had an old widows XP system in the shack for a while-then 'upgraded' when I combined my home office and shack.
Again the processing requirements are real low for most digital modes-nearly anything with a sound card/chipset will be more than adequate.