I have seen this so called "requirement" on occasion and think the company that requires it needs to go though a security audit of their own. I also think that anyone that requires Domain Admin/Root access for their software to run doesn't know a thing about security (I get this commonly too). The problem you may face will be through regulatory compliance for various things (as some here mentioned PCI already) There are other regulations you should probably try to discover if you are required to comply with especially if you will be hosting PII (Personally Identifiable Information). Not just talking HIPAA, but some e-commerce sites require a little more security for their data.
That being said, as long as there is a firewall to the internet and your server does not have critical internal only ports open, in most cases you would at least pass basic security requirements, albeit not optimally. Make sure you run a test using something like NMap! As for alternative ideas, you could set rules in the local firewall to allow unrestricted traffic between the servers involved while limiting access from all others to the default rules. You could also set up IPsec tunnels between machines to encrypt that traffic. I would at least make sure you have a path to separate as many features that require higher security as possible. For instance, PCI will require more strict rules and if that is a problem for this vendor, budget a separate server for transaction processing.
Some of those options may help find some middle ground between your requirements and theirs (after all, you are not limiting traffic of interest to them). I would push to have a Business Associates agreement signed by them saying that by not following your configuration requirements, they will hold some or all of the responsibility for a breach involving their solution. It would still hurt you in the case of a breach, but at least it puts them on edge to make sure they do everything they can to secure their solution since they now have a financial consequence for a breach.
Last but not least, make sure you put the POS system in it's own VLAN separate from all other systems. Lock down that network to only POS traffic and put no other device unrelated to the POS system on that network. This is part of the reason for recent breaches from some of the major retailers.