I absolutely must caveat this page with a general message, if your entry point into record collecting is contingent upon finding a turntable or record player for under $100, I strongly suggest saving up just a little bit longer. It's not that these record players won't suffice, it's just that the quality of the components are not likely to last in the long term and could even, in some cases, damage the vinyl you're playing. Also important, the turntables themselves aren't particularly upgradable or adjustable, so if you have a problem with one of these, you might be faced with just buying another new record player again later. My overarching suggestion is that you save up until you have just under $300 and buy this turntable, which is great quality and should last you many years of enjoyment and use.
That said, here is the best turntable under $100
Audio Technica AT-LP60 (typically $99)
The most obvious standout among low-cost budget turntables is the Audio Technica AT-LP60. It's a solid little turntable that hooks up well to a stereo in your home, or works well with the powered speakers you already have, because it has a built-in phono preamp. It doesn't get much simpler than that. This record player will play your records at 33 1/3 RPM or 45 RPM as you would expect, it's fully automatic, and affords you the essential element necessary to start listening to vinyl--a decent turntable. If you are trying to get into record collecting at the absolute lowest price possible, this is probably the most cost-conscious quality turntable you can buy before you start seeing models that will be obviously lesser in quality and could damage your vinyl.