I just saw this term on a job I was applying for and I'm not entirely certain what it means. If I had to take a guess at it I'd say it means supporting end users of other vendors software, but I'm not entirely sure about that.
To give you some context the job is at a hospital as a web developer/technical analyst.
I see that as similar to what you state but with a small caveat. I suspect it's just experience with application support with a vendor (not another vendor).
In other words, they're looking for the fact that you worked for Dodgy Brothers Software Ltd in supporting their top-of-the-line application, BCPL Toolkit.
You'll probably be quizzed about how customers raised faults, how they were prioritised, how they were classified as bugs or feature requests, how they were tracked, developed, tested, deployed and so forth.
It sounds like they want you to be a pseudo-third-party product provider at this hospital, not a bad idea since it will probably provide a fair bit of autonomy (it would be unusual to treat you as "separate" while still maintaining a tight control on how you do your job. Of course, like any vendor-customer relationship, they still control the purse-strings so autonomy is a two-edged sword.
The other possibility is that they're just looking for experience with dealing with vendors (if they want you to evaluate/integrate real third-party products).
But I see that as less likely since that would less call for "Vendor Application Support" and more call for tendering, requirements and evaluation skills which you don't mention.
And remember that almost any experience can be subtly morphed into exactly what the employer requires :-)