Finding the impedance of a transmission line
Does this method make sense? 


Yes. This is called time-domain reflectometry (TDR). You can buy attachments for certain oscilloscopes that do this, or dedicated TDR instruments.


  If not what are possible problems I might face? 


In order to get a good result (one that can distinguish important features in the transmission structure), your DAC (or other kind of function generator) must generate a very fast rising edge. On the order of 100 ps is probably a good starting point, unless you have a very long transmission line (10's of meters, at least) and are only interested in gross structures of the line (is it terminated with a short or an open, for example). 

Similarly your ADC must have fairly high bandwidth (100's of MHz at least, GHz preferred) and corresponding sampling rate for the results to be informative.


  I'm wondering if I would even be able to see the reflection, or if it might be too small.


You will be able to tell the difference between short and open termination even with a very simple set-up. Measuring, for example, the propagation delay through the line will require more care, and the precision of the result will depend on the bandwidth of your measurement system.