I really don't want to have to purchase a huge, loud 4 channel amp if I don't have to.

You don't have to. Just get a decent-sized, medium-volume 4-channel amp of decent quality like this Jensen XA4100. Only $159.00 at the moment. (Ignore the fact that it comes with a promo pair of cheap subwoofer drivers and just use the amp.)

Remember that a powerful amp does not have to mean "loud". The idea with amps is to have enough headroom so that you can send it a very clean signal at moderate listening levels.

Is it not possible to simply wire left/right channels from the amp in coordination with the outputs of the RioCar?

You are talking about driving two speakers per amp channel, so that you can run four speakers off of two channels. This is called wiring the speakers in either "Parallel" or "Series" mode.

This can be done, but your amplifier has to be able to handle it (meaning you need a more expensive amp to begin with).

See, when you do that, you are either doubling (series) or halving (parallel) the reistance on that amp channel. This is called the "Load" that the connected speaker(s) present to the amplifier. In most car systems, the speakers and the amplifiers are matched to a given load (usually 8 ohms if I recall correctly), expecting that only one speaker will be hooked up to a given amp channel.

So the speakers will either play twice as loud or half as loud, depending on how you wire them. If they're playing twice as loud as the amp was meant to handle, then that puts a strain on the amp and makes it run hot. If they're playing half as loud, then you have to turn the volume way way up to hear them and you get distortion.

Some amplifiers (as I said, the more expensive ones) will say things like "stable down to a 2-ohm load". This means they're meant to tolerate that kind of abuse.

But it's much easier and better to simply buy a four-channel amp. It need not be huge or loud, just good. And remember that they have gain adjustments so that you can set them to never overdrive your speakers (in fact this adjustment is a critical part of installation as described in the FAQ, here).

Let us know what you end up doing.