I put a recorder app on my phone and when I was in for one of my regular appointments I asked my doctor if I could record the heartbeat andhe was happy to let me.
I'm not sure about making a tape or recording, I don't believe your doctors office would/could do that? I do know my mom got me a "heartbeat bear" which has a small heart-shaped recording device which you record the heartbeat on during the ultrasound or when they use the doppler on you, and you can lock it in place and save their heartbeat in a bear/stuffed animal.