I checked the other forums, but couldn't find a similar problem...

Anyway, I played the game once and it worked fine, no problems whatsoever. When I tried playing another game, a creepy looking black laser-beam type thing (for lack of a better word) shoots down from the sky onto the field. The game will show the anthems and all that, but after that it'll crash and go back to windows. So far I've had no luck solving this problem

I have a similar problem with FIFA 2004, EA told me that it's a problem with my video card, but I've been able to work around it because when the black laser shows up, I quit the game, start it up again, and problem solved.

Does anyone have any suggestions? Hopefully any suggestions short of me having to buy a new video card?

You won't have to buy another card.
First search for latest driver for video card, because that is the most likely cause of the problem.Also download the latest released direct x drivers.

Pardon my ignorance, but where would I search for one of these drivers? What exactly do I look for?

depending on what videocard you have , you will have to search for drivers.
Please check brand and type of your card and I will point you in the correct direction

Name: SiS 650_740
Manufacturer: SiS
Chip type: SiS 740 Rev 00

OK , it is clear.
This video card is integrated on your motherboard.
I am afraid the possibilities are limited.
For dirvers you can check on :www.video-drivers.com
or http://list.driverguide.com/.
But knowing now that you have an integrated video card , the best advise I can give you to buy a Geforce card or ATI and disable the on-board card.
Sorry that I will have no better solution.