hy - sorry to tell you what you already knew. I didn't understand your purpose, and I've never heard of vibration monitoring used in this manner, although there must be something to it if the manufacturers are talking about it.
From what little I know, the primary factors involved in transformer winding loosening have historically been thought to be through-faults, and thermal aging of the insulation. Related tests as you mention are single-phase leakage reactance and excitationi current and internal inspection. Recently there have been a lot of papers on Frequency Response Analysis for detecting winding movement. I'm certainly not trying to dispute the importance of vibration... I hadn't heard any discussion on that aspect before, but it makes sense that it could be another factor in winding loosening. Also I have never heard of fatigue failures of connections within a transformer.
At our power plant we use Entek IRD Datapak together with desktop software for vibration monitoring of rotating equipment. There is a range of vib monitoring equipment available starting with "vibration pens" sold by skf, giving magnitude only. You can add capability to provide spectrum display, then capability to interface with pc program, do multi-channel analysis, balancing etc). If you have any friends in a power plant or industrial plant with rotating equipment, they will have specialists in vibration monitoring. Vendors include Rockwell/Entek, SKF, CSI and a few others.
One other story that comes to mind on the subject of transformer vibration (not related to your question) ... I heard a report where a generator stepup transformer had a fan blade which apparently developed a crack which went unnoticed.... resulted in increasing imbalance and vibration until the fan blade finally fell apart... shrapnel cut into the cooler tubes resulting, in sucking air into the transformer (cooler was under suction), caused relief valve to lift, and then operators took unit offline in response. Other fan blades showed cracks near the bases of the blade. I have seen similar cracks myself but never a catastrophic failure. It certainly seems like on-line vib monitoring would detect the cracked fan blade as evidenced by vibration at the rotating speed of the fan.