Sometimes the causes of tuner/amp vibrations can be traced to the battery compartment or the wiring loom. The battery can set up a buzz if it is loose in its housing. If it is a typical 9V, few wraps of electrical tape around the girth of the battery to make it a light press fit into the housing is usually enough to eliminate this problem. Anywhere the wiring loops contact the inside of the body without being secured is another potential buzzing source. Use the 'chopstick' method (through the sound hole) to move the wires around.
If chasing buzzes/vibrations, it is best to adhere to a systematic check list.
This is how I go about it:
Start at the top and work towards the tail.
Clip on electronic tuner mountings are notorious for buzzing. Temporarily remove the tuner and pluck the offending string/s.
Some geared tuning machines (usually cheaply made ones) can rattle and buzz. Sounds unlikely, but this was on one occasion a most vexing problem to track down.
Poorly fitting, loose, unglued nuts held in place only by string tension, may buzz, but this is rare. One or two sparing drops of glue will fix it.
High frets, low string action, and incorrect neck set are bigger issues that can be referred to a luthier for correction.
'Dry' glue joints between the fret board and sound board can buzz. This may be very difficult to detect. Flex the neck and look for a gap. Some makers do not glue this joint so seeing a gap does not mean that it is a definite problem. Exerting downward pressure over the joint with the end of a ruler while plucking the strings will clarify things.
Wound strings that are starting to unravel can be problematic. This is more common with steel strings.
Poorly fitting, loose saddles can tilt forward in the slot producing sketchy contact. Pack the gap or better still, make a new saddle with a firm sliding fit.
Loose neck fixing screws and/or washers. No need to elaborate on this. It is usually caused by age shrinkage or dehydration of the neck block.
Loose, split or cracked braces. These can be hard to identify and difficult to fix, best left to a repairer or luthier.
Last but not least, loose jack plug nuts and loose metal strap buttons.
Most of these items were gleaned from guitar, but are also applicable to ukulele.
I'm sure to have omitted something, so chip in if you can add to this.