Can I get three cheap Windows XP copies and just upgrade them for free?
As already noted elsewhere: no. Windows XP doesn't qualify for the upgrade.
Or is the upgrade restricted for copies that are sold before a special date?
This isn't true at all. The famous free upgrade is restricted for copies of the Windows 10 upgrade that are INSTALLED before a special date. You can't just own a Windows 7 license. You need to go through the installation process, which involves activating Windows 10 before the date. Once you do that, if you want to downgrade back to Windows 7, you can do so and you're still licensed to upgrade back to Windows 10 later, if you like. However, getting the Windows 10 license requires more than just purchasing the Win7 license. In exchange for the free upgrade, Microsoft requires that people go through the effort of getting the PC off of the old version of Microsoft Windows before the deadline. (They are really wanting people to get Windows 10 installed, and actually experience Windows 10.)
Which alternatives I've got to get Windows for these 3 PC's?
Consider getting Windows pre-installed on brand new PCs.
I decided to just use one vendor (New Egg) to show some quick examples:
Granted, I chose a computer with some rather crummy specs, but if the computers that you're currently using are old, then some of the cheaper computers being sold may be an upgrade. (Especially if you're willing to get Refurbished.)
Also check out Microsoft BizSpark for Microsoft software (if you qualify for that). As a side note, also check out TechSoup.org for any other software needs (if you are a non-profit, and qualify for that).
Note: None of the above details are meant as recommendations to proceed with. I'm just trying to throw out some ideas to open your eyes to some options, and letting you know of some requirements I have been made aware of.
Personally, I like Hastur's answer: checking out various versions of Linux (including Kubuntu) may be better. Since there are concerns about people not being tech savvy enough for Linux, they may often just need a graphical interface with a web browser: That was the main basis of a lot of Google's marketing for Chromebooks. However, you did ask for "alternatives" ... "to get Windows", which I interpreted as options that did involve getting Windows (not alternatives to doing that). My choice of New Egg was just that I felt confident that I could find prices quickly so I could make this answer quickly, not because I'm trying to encourage usage of them over any other vendor.
Also, if you're looking to save money, consider the Invisible Hand and PriceBlink browser plug-ins, which I have liked (and do recommend... okay, I am making a recommendation here), and maybe the PriceTrace Toolbar (which I learned about later).