try the "I don't have one, I only have a drivers licence gambit".

No, that won't work. There's a lot going on in a situation like this, and I'm not sure if I can convey what I want to say clearly here, but, I'll try.

First, it should be illegal for an internet company to request your zairyuu card. It is fully legal for you to show it to them voluntarily, but they shouldn't be able to ask for it. (Again, IANAL, but that's my understanding.)

Second, once someone at a business gets it into his head that he needs to see your zairyuu card, he will reflexively insist that it is the rule. It comes down to, "This is the way we've always done it, so it must be true." He isn't going to voluntarily check the rules for you - he's going to sit there, stubborn as a rock, and say, "Muri, muri, muri, I must see your zairyuu card and nothing else."

I've run into this in a few different situations, and every single time I have pushed the person to check the rules. And when the staff member checks, they always find that the zairyuu card isn't, and never was, necessary for us to complete our transaction. You can do this politely - just say, "I'm pretty sure we can do it differently; please check." Most of the time, there will be an alternate method that the person didn't know about.

In other words, no. Saying you don't have a zairyuu card plays into their hand. (Hell, they might even report you to the police). They'll just shrug and say, "Then this transaction is impossible. Have a good day." Most people here really aren't going to go out of their way to solve a problem for you unless you ask them to.

This isn't me being a bitter expat - as I understand it, there is a very good reason that Japanese people tend to do exactly the thing you ask them to do and nothing else - but that's what happens in these situations. Unless you directly ask for an alternate method, they won't offer it.

Tl:dr, don't lie that you don't have a zairyuu card - ask them to check the rules for you.