Hardware is cheap and getting cheaper. The systme may not have to last 15 years. Some important questions that need to be answered really:
Are you going to continue using the existing software? If so then you need to worry more about weather ANY Microsoft OS will still be able to run that software in 15 years (you said the existing software was on Win95). If you are going to run that same software, I would not bank on it running on Microsoft's current OS in 2024, which means that you really DO want to install an OS on the box and have it continue to run for 15 years as a replacement box down the road may not be compatible with the legacy OS you are using for the program. If you are writing custom code, then you can make it web based, in which case, the OS is irrelevant as long as there will be continued versions of the database you use and as long as you design with standards in mind, avoiding Flash and other things that will likely not be compatible in 15 years.
Can the software run under WINE or DOSBOX. If so, then you are free of the OS constraint. Can you virtualize? If so, then you might want to choose an exisitng VM solution (perhaps VMWare, or something simialr) install whatever OS you need to run your software on a VM and keep backups of the VM. When the hardware dies, buy something new and cheap, install the current version of the virtualization software, and copy ni the latest version of the VM and you are good. The drawback here is that you are banking on any of the existing virtualization solutions to either still be compatible in 15 years or to provide a migration path. If you go with a VM solution, and the software changes format at some point, providing an upgrade for you VM, you HAVE to do it, none of the "if it ain't broke" argument because if they switch from version 1 to version 2 in 5 years, and from version 2 to version 3 5 years later, they may provide you with an upgrade path from 1 to 2 and one from 2 to 3, but not likely from 1 to 3, and don't bank on that upgrade from 1 to2 still being available when 3 comes out. You don't want to be stuck. VMWare is popular and has provided migration from early version of the format to newer. I think either they, or a compatible F/OSS version will likely be around in 15 years, but I am of course, not a fortune teller.
How much are you willing to lay out? If they use 2 workstations, you could theoretically buy 6 identical $200 systems (minus the keyboard and mouse), expecting each system to last only 5 years. As long as the rest of the hardware is identical, a simple disk image could make deployment easy. As long as you keep backups of the data then migrating from the original hardware to the backup hardware is a simple backup restore and you can be up and running again in a few hours)