Can I use a Zener diode to convert 14 V into 5 V in a small-current circuit?
I would personally use an 5V LDO regulator but as OP requested it is possible using a Zener diode, in my example I used a 1N751 5.1V Zener diode. As @mkeith said you will need to limit the current. 

Max current is calculated as Power Dissipation/Voltage, in our case it is 500mW max/5V giving the maximum current as 100mA although the datasheet also states that the maximum current is 70mA so we will stick to that.

Rs = (Vs - Vz)/Il = (14.6v - 5.1v)/70mA = 135 Ohms 
E12 series makes that 150 Ohms

Assuming a 1K load
Il = Vz/Rl = 5.1v/1k = 5.1mA

Max zener current
Iz = Is - Il = 70mA - 5.1mA = 64.9mA



R1 is Rs
R2 is the Load