That is essentially what I was hired out of school to do. I walked into a massive heap of ASP.NET (with VB) and MSSQL stored procedures that didn't really work at all, and had been through the wringer of a few cut-rate outsourcing groups. I struggled along with it for a few months figuring out how it was supposed to work and trying to duck-tape it together, doing a lot of support with customers that were trying to use it, and talking to them about what they were trying to do.

Then eventually I decided I wanted to learn some newer tech, so I started playing with Linq-to-SQL and ASP.NET MVC and Razor and Bootstrap, and over the course of three or four weekends and evenings I did a ground-up rewrite of the whole thing for fun. After a a bit more time flailing away with the old mess, I showed my side-version of it off to my boss, and it wasn't that hard a sell, being much prettier and less buggy.

It helped that there was nobody around who was invested in the old code base.

Generally, I've found it is a lot easier to effect this kind of change if you just do it stealthily and present it as a fait-accompli, because otherwise people get so bogged down in debate and fear that any impetus to actually take a risk and do something evaporates.