ChunkyZergling
Super Freak
If you're into Ikea hacking, take careful note of the dimensions and then do a glass search on their web site. You can even look at Pax glass doors, etc. Just keep in mind that you can't cut or otherwise modify (drill, etc) any Ikea glass as it's all tempered. Ultimately you might even consider swapping out the Besta for something else that has a mating glass door from them.
Keep looking at different glass suppliers. For $500 I can buy glass for a complete shower, 3/8" thick and tempered - at least 1 wall, probably two, and 1 door plus fittings and silicone.
BTW, you might want to check Home Depot in your area to see if they sell glass.
I just purchased two regular float/annealed glass panels, 1/4" thick, 30" x 64" with four polished edges for $75 Canadian each (North of Toronto in Vaughan). It's a 40 minute drive. A more local supplier (West of Toronto) would probably be $300 for both.
Also, don't leave it up to the glass cutter to tell you what you need. Don't even tell them what it's for. Just specify the dimensions and properties you're looking for. If you're actually *in* Manhattan, try somewhere in the suburbs if you have to. Or in Jersey.
You won't need more than 1/4. 1/8" might be fine. You also don't need it to be tempered (though it's stronger and safer in case of breakage). You do need all four sides of each piece to be polished to prevent a cutting hazard. You have various options for hinges - pre-drill the glass or use slide-in hinges like many of the lower-priced Ikea cabinets. These can be bought from Home Depot and online on eBay for pretty cheap (especially if you need a bunch of them. Chrome, black, white, brass, whatever floats your boat. You can pick up suitable slip-on locks on eBay as well. No more than $5 ea.
Acrylic/Perspect/Acrylite/Lexan and other polymers are generally going to cost you much more than an equivalent sized piece of glass, for both width/length as well as thickness, even if you go to a decent plastics supplier.
Thanks! Good ideas here. Also drilling the holes in back. I have a Home Depot 5 minutes away from me. What should I ask them about cutting or whatever for hinges? I don't know how to attach the glass.