Yes, it is possible to audio quantize in Reaper and all popular DAWs.
In general you'll find two different approaches to audio quantize (or a combination of): with or without time stretching. The best results will depend in the material you are modifying. Without time stretching (which is nothing more than cutting a piece of audio and putting it somewhere else) tends to work better with percussions. Time stretching works better with sounds with larger decay and/or sustain, since no time stretching would leave audible cuts.
You can do both approaches manually (and some people do!), so the question is if your DAW has tools to automate the process, and which approaches are available.
In Reaper you use a combination of Dynamic Split and Quantize tools. You first split the audio into sections based on its transients, and then align them in relation to a time grid. This is a no time stretch approach.
Seems that it has its issues, though. From http://wiki.cockos.com/wiki/index.php/Audio_Quantization_in_Reaper
Natively Reaper has rather poorly implemented audio quantization functionality. However when used in conjunction with the recommended tools below, you'll get a fully functional way of doing this.
I haven't worked a lot with Reaper, so I don't know if it has an automated time-stretch-based quantize tool or how good its time stretching algorithm(s) is(are).
Other DAWs offer other alternatives/approaches. Pro Tools' Beat Detective and Logic's Flex Time are some examples.
Some notes if you are still in the "choosing a DAW" stage.
As time stretching affects the timbre of the source, an important detail when choosing a DAW is its time stretching algorithms (which can be used to quantize, among other things). How many algorithms can I choose from? How good are those algorithms? How much control do I have over them? Can I time stretch quantize without noticeably affecting the source's timbre? To which extent (since the larger the stretch, the larger the timbre change)?