Most DAW programs don't enable MIDI THRU by default because in some setups it causes the notes to get doubled depending on how things are set up. It's something you'd have to turn on yourself.

Look for an option in Reaper to enable MIDI THRU on the DAW, or perhaps just the track itself. I don't know where it is exactly because I haven't ever tried to do MIDI recording in Reaper. There may be additional complexity there, such as a way for Reaper to filter or modify the MIDI that's being passed thru, such as passing through only notes but not MIDI controller messages, or changing the overall velocity curve before it passes the notes through, or changing the MIDI Channel that the data is being passed through to, or that sort of thing.

Perhaps something in here: http://wiki.cockos.com/wiki/index.php/Recording_MIDI

Something else worth trying: Instead of using an external synth, try one of the plethora of virtual instrument plugins that can be used directly in the DAW. Reaper may even come with some. You won't have to enable MIDI Thru to get those to work.