For me it's too broad.
As for me such question fall into the answer to use processmonitor if you can't contact the software vendor.
Does we support application not supported by the vendor/coder, I don't think.
The OP told that the vendor just can't find the problem, then the vendor got a bad coder, as that mean that app is badly coded and they doesn't even know why it need admin right. Helping the OP to get a processmonitor's dump will someday give a result, that is maybe bypassable or that would give enough information for the OP to escalate to the vendor, but it's for me a break and fix, not an actual fix for the problem as we do the vendor's work there.
I give an actual case I lived. I escaladed a bug on a windows software, citrix desktop delivery agent in my case. The vendor asked me a processmonitor on boot, sent it via ftp, he readed it and offered me a fix. This is how it should work in my opinion.
Edited : My opinion told is based on the fact that the OP already got an answer from the vendor. Not like hes left in the wild without answer.