I'll admit to bias on this subject -- I work with one of the larger FileMaker development shops out there, and have written the odd book on the subject. We actually employ many respectable developers who love using FMP. I'll try to keep it brief. :-)
FileMaker Pro is a rapid app development tool. It's primarily client-server, though it has some very respectable web publishing capabilities which work well for many applications. It is not SQL-based, but does have ODBC and JDBC interfaces, as well as an XML/HTTP interface.
As far as lock-in, FileMaker Inc has grown sales steadily, with very significant growth in new users who are attracted to the platform's solidity and ease of use.
I think Matt Haughton nailed it -- for the right applications, FMP is simply the best choice going. That said, your customer is looking at apps written in FMP Pro, and you need to evaluate those apps on their own merit. They may be good instances of FMP development, or they may not.
To know more about FMP's fitness for the task, we'd need to hear more about the proposed application and user base. Are these indeed web apps, or client-server? How many users will be using it? Do they work at one or two site, or are they spread across the Internet?
Happy to elaborate further if there's more interest.