Without knowing exactly what you mean by "track their entire business process" the answer is almost certainly 'Yes', as lots of companies use Filemaker systems for that purpose. More specifically:
'The application will have approximately 5 simultaneous MAC users' - peer-to-peer will handle up to 9 simultaneous clients. If you exceed that then bolting on Filemaker Server raises that limit significantly.
'Is it possible to do the programming in FileMaker Pro 12 on my Windows 7 laptop' - yes: it is truly cross-platform. Well, very-nearly. There are a couple of script steps that are unique to each platform, and you need to make sure the fonts you use display correctly on both platforms (Mac renditions tend to be ever so slightly larger, so leave a little breating space), but those are very minor issues.
'...with the data housed in MySQL 5.1 Community Edition' - from FM's point of view there is no reason not to hold the data in FM, but maybe you have another reason to use MySQL. From memory you can 'mount' the MySQL table in the Relationships Graph just as if it was a native Filemaker Table. You can test that easily with the 30-day trial.
'put it on a Windows 7 PC to be the host and share it peer-to-peer to MAC laptops on the same network as the host PC' -no problem.
BTW: you don't actually need to buy the Advanced version to do the programming or sharing. The user experience is exactly the same as with 'standard'. The benefit with Advanced is to the developer - there are tools there that usually justify the extra cost.