I would say these are average numbers on time it takes to get a good vibrato going.
Slow/Proper practice everyday with drills – 5 Weeks
Random vibrato in songs – 6 Months
You can see that you can learn it a lot faster if you apply slow and proper practice to learn it. Many people don’t do this, and it results in then never really getting it. You can actually form a lot of bad habits with vibrato, so it’s important to learn it the right way. I highly recommend following this vibrato resource to learn how to do it properly.