Is vibrato gifted or is it developed by constant practice? Are there any exercises or techniques to develop a good vibrato in singing?
From a personal perspective, my vibrato started coming out after learning good breath support from the diaghram. If I kept my throat relaxed and held a note in a comfortable area of my voice range, the vibrato would kick in.
It almost felt like hitting a sweet spot - a balance between diaghram control and vocal chord control. Once I could evoke it reliably, I was able to start controlling it.
For the specifics on how to actually start the process, step-by-step, you might want to read this on Wiki-how:
Best of luck!
Vibrato is your friend. All singers, in any style of music, could benefit from learning to use a little vibrato because singing with vibrato is less fatiguing on the voice than singing with nothing but a "straight tone". When singers suffer vocal damage, it's often because they have sung for years with no vibrato and don't know how to use vibrato and control it.
When they start out, in their untrained voice, most singers, men and women, find themselves with very little vibrato. They need to develop vibrato (and how to control it) through practice. Any good voice teacher can work with you on this. It's fundamental to singing lessons. When you learn to produce and control vibrato, you can decide when and when not to use it, and how much, depending on the style of music you are singing.
For some reason I was the opposite: my natural untrained voice when I was young had a great deal of uncontrolled, wobbly vibrato. My voice teachers (I started studying voice at the age of 22 years) had to do a little work to show me how to reduce the vibrato and control the degree of vibrato. It's actually always been harder for me to sing pop or jazz (rather than classical or opera) because pop and jazz are supposed to be sung with very little vibrato.