And that would be a valid question, as 1996-you would be using Netscape Navigator 2 or Internet Explorer 2. Good luck building web apps for those browsers ;)
Correct me if I'm wrong but I don't believe client-server applications were a new idea in 1996. Instead of having a fat client, generate HTML and go back to the HTTP server for every request. The application wouldn't be fast and the code probably wouldn't be pretty, but it would definitely be a software application that runs purely on the web.
There were few web applications back in 1996. I can only think of two major ones: Lotus had a webmail product called cc:Mail and eBay just started offering online auctions.
Back then, you wouldn't use PHP, CSS, Javascript, XMLHttpRequest or Flash. They didn't exist or hadn't matured yet.
Most likely, you would've used a Java based solution, such as NeXT WebObjects. That's what e-commerce businesses did back then. However, you wouldn't have developed the product on your own. You would've needed a team and plenty of funding, in part because WebObjects came with a $50,000 price tag.
I encourage you to read this WebObjects advertorial from april 1997, courtesy of the Wayback Machine. It's a good illustration of what was considered modern web development:
You would have used Perl or PHP - I was using both from early 1996, and they worked just fine. Yes, they hadn't "matured", but they were functional well before Java or ASP were even available. Server-side Java was not something as widely available as Perl or PHP, and Java as a language was not written for the web like PHP was (easily embeddable in HTML, straightforward DB access, etc).
WebObjects - didn't have any direct experience with that back then, but it and many other proprietary options were quite pricey, IIRC.
I assume you mean something like Google Doc's spreadsheet app. Let's say it's 1996.
The DOM wasn't standard, HTML4 was still in the works. I'm not sure if IE2/3 or Netscape 2 could even dynamically access table elements at the time.
Javascript performance was utterly terrible, and JScript had a strong presence that created an important and problematic language/support divide.
CSS didn't exist at all, your visual styles were either images or elements rendered as the browser saw fit.
XHR didn't exist and wouldn't exist until IE5, 4 years later. There was no way to update a page without a round trip to the server. This was also during the days of AOL and Dialup.
I envision a table filled with a grid text input boxes and some buttons to add more rows or columns and maybe to do a few other things. We'll do things server-side if we have to.
Ugly as sin? Slow as hell? Absolutely. Probably doable, though.