Many scrapbooking stores have die cut machines for their customers to use for free, but I've never had to cut out several hundred of something. If you don't do any scrapbooking yourself, don't go to the expense of buying a die cut machine, and then buying a die for the machine. Most home use machines require you use their brand of dies, so making your own wouldn't probably be an option. Basically, the way it works is the die is a shape that's in a rubber stamp type block, and there's blades in the shape of whatever shape you're cutting. I've used the die cut machine at school, and I have to trim the paper to cut several at once. Depending on the machine and the thickness of paper you're using, you can load 2-3 thicknesses of paper at once, but since you're using a heavier weight paper, I wouldn't. I would do it one at a time. The size of paper you can use depends on the size die you have as well. If your die is 4"X6", you probably have to cut the paper to that size (a little bigger) to cut out the die.
Your best bet may be hiring it out. Call around to places like Office Depot, Kinkos, any stationery store, etc. See if they handle something like this, and how much they would charge. It might be worth it considering you're having to do several hundred of them.
Also, if this is for a school thing, check at school...they do stuff like this ALL the time. In our school district you have to send something like this to the district's printing facility, but it only takes a couple of days, max.
Good luck!