It depends on what you mean by feeding your website. You could use a hosting company for your FileMaker data and create data pages. Then you can reference these into your website pages. You may be able to get your data into MySQL, but it will require perhaps a nightly batch upload via something like FTP or another method.
The easier route would indeed be either move the entire solution to MySQL or move the website to another hostthat supports FileMaker.
I have done as suggested and now I have my website hosted with one hosting provider while the database and data pages are hosted with the filemaker provider and I am calling the datapages with a url -
I work with the database on daily basis as the database constitutes the back end of my business - However I work on the database locally as remotely is very slow to work on.
Being this the scenario would you recommend me hosting the database and datapages at my office by purchasing the filemaker server?? Is it difficult to manage. I am computer literate but no php experience.
It depend on how many people access your system. The problem of hosting things locally is the cost of bandwidth is actually fairly expensive. You can expect around $100-$150/Mb for a business line on coax. The minimum T-1 (1.544Mb) is usually around $350-$600 depending on where you are. A T1 would be about the absolute minimal required if you consider hosting anything.
Then again, at that price range, it also become feasible to buy a server and colocate it at a data center.
You can also get a Windows dedicated servers that's pretty decently specced for about $150-200 and host your FM and web server on it. And it'll be fairly decent if you Remote Desktop into the machine to work on your FM.