Does my Toshiba 37HLV66 LCD TV/DVD Combination have the capacity to properly accommodate a Blu-Ray player? Some technicians I have asked say, "yes" and others say "no."

This can be both yes and no. Looking at the specs of your TV it would not be displaying in full 1080p (The highest resolution for HD)
The best part of Blu-Ray is viewing this in 1080p. Your TV can accommodate a BD player and would be best set up using the HDMI input port on your TV. But, it will limited to your resolution of 720p or 1366 x 768 on your TV and most DVD players can upscale the quality of a movie to 720p or higher. It would be worth getting a TV that has 1080p to go with a Blu-Ray Player.