After getting a Vassal you will have to wait 10 years and after those 10 years following must be true.
You need at least 190 in relationship with your vassal. (Increased by royal marriage, and diplomatic relationships and gifts. Vassals of other religions is much harder to get this high with, so those are easier to conquer, rather then Diplo-annex.
You also need to be at peace
The vassal must be small enough compare to you preferable just a few provinces. or they won't be interested in joining your country.
When all of these are true send a diplomat to your vassal and start the annexation.
During the annexation process you may enter a war, However if you do make sure you protect your vassal provinces, if they are occupied, it endangers the annexation process and you might have to restart the whole process.
Side note after you annex a vassal you will get a cool down period when other vassals are less likely to allow themselves to be annex so it can be a good idea to group them into the same annex process. ie start their annexation at the same time. Also keep in mind that any army/ship own by the vassals will becomes yours, which may take you over the country maximum and/or cause havoc in your economy.