I doubt very much whether a card is needed for a corporate rate. (I am assuming that your DH's company has negotiated special rates with National for their employees' use). I've never had to show anything when using my company's contract.
I signed up for the Emerald Club via BizRate - some type of online organization. I have never had to show anything for that either.
That said, I've always been asked for my Disney Club card if I used it to obtain a discount. I've also had to provide an actual coupon if I used one to get a discount. If I use the EA or EC, I have always given them my Emerald Card, so I don't know for sure if they would ask for it if I didn't volunteer it.
In summary, do not worry about any card for your husband's work code. You should not have any problem at all with that. If you are worried, have you husband take his company work ID with him. At the most, that is all you should ever need (and I doubt you will need that).
Enjoy your trip!