What Budget Video Card is Recommended for Solidworks?

An SSD would definitely improve performance.  However, Intel HD graphics are not even supported.  Depending on what you are doing with SW, and since you are teaching K12 I'm assuming you aren't doing complicated assemblies or parts, a basic supported video card may be sufficient.  And I'd get that first.  If you aren't running supported hardware, strange things can happen.  If you are using Core 2's then you probably won't benefit from a newer faster card on the newer PCI bus.  So, you may even consider a used card, say a quadro k600.

Dell outlet has some pretty good prices on newer used computers that come with a factory warranty.  Do you think your administrator would consider a new computer?  Take for instance this PC: Dell Official Outlet Refurbished Laptops, Desktops, Servers & Monitors.  It is $567 and comes with a 128GB SSD, Firepro 2100 (SW supported video card), 8GB Ram (upgradeable at a later date), 3.9 GHz-i3 (SW really benefits from clockspeed over cores).

Have a chat with Dell and give all specification of the current system. They will give you a list of video cards compatible with the system.