Well, that budget kind of restricts you to entry-level, but you can still make some nice-looking videos with relatively inexpensive equipment.
A GoPro will go a long way. I recommend at least a Hero3+ Black or Hero4 Silver. I started PeachHack with a Hero4 Silver and it has worked great. If you set the FoV to “Narrow” there is very little “warping”, and I use Premiere to remove the last little bit of fisheye, but even that much is probably not 100% necessary. (I now use a Panasonic HC-V770 for recordings.)
The big thing that most people forget, though, is lighting. Lighting makes a gigantic difference in video quality, partially because it brings out the color of the cards more, but also because it makes it so the camera doesn’t have to run in low-light mode, which would make the video grainy. I started out with just some simple work lights from Home Depot with Daylight Flood CFLs, and then later upgraded to a nicer lighting kit I got off Amazon for ~$70.
Another thing to think about regarding lighting is reflections. Try to grab a spot in the store where there aren’t any lights that would directly reflect off the cards, if possible. For example, you could try setting up at a table that has one end up against a wall, and set up the camera facing the wall, angled down at the table. A circular polarizing filter could also help, but if you set up in a place with no reflections, you can probably skip this part.
You’ll want a nice, simple camera tripod. Make sure you put the tripod on the table when you record, rather than next to the table, so that people don’t accidentally kick it. If you get a GoPro, you will also need a GoPro tripod mount.
Also make sure you get a few large MicroSD cards. A single 64GB microSD card can hold ~4 hours of 1080p30 video from a GoPro Hero4 Silver, so 2 of them would get me through a tournament that was 5 rounds of swiss + cut to 8. Anything longer, and I would want to get another.
As for power, I just recommend getting an A/C adaptor for the USB plug and keep the thing plugged into an outlet. You’ll probably already have a power strip for the lights, so just plug the camera into there.