What could cause a strange vibration from the engine on a 1999 VW Beetle?

Unfortunately, there are too many moving parts on a vehicle to give you an adequate answer. It all depends on the circumstances under which the vibration occurs. (Such as it only occurs while idling, accelerating, etc...) I can tell you that there are repair shops that will give your vehicle a free inspection. Just call around. Even if there are no mechanics willing to look at it for free, I'd still get it looked at. Unusual/abnormal vibrations can cause damage to perfectly fine parts of your car if you leave it go for too long, thus adding to the repair bill. Sorry I couldn't be of more help.