Integration with your CRM/Contact Management system is the thing. Last place I was at used a commercial version of SugarCRM and a dedicated card scanner to feed the card module (I can get the details but I fancy most/many multi-function printer-scanners have some provision for card scanning and some of those are cheaper than buying replacement print cartridges, certainly cheaper than a purpose scanner though you get what you pay for to some extent).
Does your CRM system make provision for cards? Or is there an available module? That's the question. Really, it's the 'only' way to integrate the data. If it doesn't, maybe you need a new CRM. Your chance for revenge. 'Sales' hate changing CRMs

(the reassurances that the new one will effortlessly assume the data and functionality of the old one - and Outlook - is always a cruel hoax). But there's probably an add-on module available. Maybe you're only using Outlook for CRM? Well, that can work, there's the vCard functionality and (third party) modules to add scanned cards to contacts, I think.
I'm faintly amazed physical business cards are still so important (so 19th century). But Sales are right. The contacts are critical. So, what are you using for CRM/contact management? LAN or VPN could come into it too, I don't know.