Train stations and Versailles

I don't know where you're trying to book your ticket, but you should be able to choose either station. That said, both stations are fairly close in and I don't think either would be a problem.

To get to Versailles Rive Gauche station, you don't take a "booked" train. You take the Train C (formerly called the RER C, and at least in May 2018 the signs still had the old name). These are like the Metro, except they go to the suburbs like Versailles. No pre-booking is necessary or even possible.

The other Versailles stations (Versailles Rive Droite and Versailles Chantiers) are reached by SNCF instead of RER. These stations are a bit farther from the chateau than the Rive Gauche station, but there is bus service if you're not up for the walk.

You can see the locations on Google Maps.

Does the RER C go from Chartres to Versailles?

The RER is a Paris commuter network, it does not go as far out as Chartres. One of the branches of line C ends at is Versailles-Château-Rive-Gauche. There are other stations in Versailles, make sure you get the correct one. The "Château" was added to the name for the benefit of tourists.
Versailles-Chantiers is a station on the national rail network (SNCF), which is why it is showing up on rail searches.

Trains from Chartres go to Paris Montparnasse station. You then need to take the métro, and change to RER line C.
I doubt you could get through tickets (may be wrong), you probably need one ticket to Paris Montparnasse, and another from there to Versailles.

Versailles Chantiers is on the line from Paris Montparnasse to Chartres, so yes, you can get a train from Chartres to Versailles. It's not the RER, though, but the medium-distance SNCF service.

Are there taxis at Chantiers station?

There is a taxi stand outside the Chantiers station, but there were no taxis there when google street view snapped the picture in 2015.

To put things in perspective, Chantiers station is a 1-mile walk to the Palace entrance. Rive Gauche station is exactly in between, so even if you were to arrive at Rive Gauche, you still have a 1/2 mile walk to the Palace entrance.

Some of the answers are confusing or missed that you will be coming from Chartres to Versailles. You will take the SNCF train to Versailles Chantiers station. There are no other train options worth considering.

From the Chantiers station is not that far, about a 15 minute walk to the Rive Gauche station where normally tourists arrive at (from there another circa 15 minute walk). It is a direct street straight out the station, so don't worry about getting lost.

There should be taxis if you are unable to walk that far, though be aware that visiting Versailles means A LOT of walking no matter what.