CDR is different program, called Corel Draw - you'd need that software to make the file.

However, very bad for printers only to accept this file - they should be versatile and it's unusual that they can't accept an Illustrator file.

I think you should talk to the printer about alternatives, like a PDF for example.

Only cdr? That's ridiculous. Corel Draw can import ai files. Perhaps they have an older version of Corel Draw that does not import the latest ai format, in which case when saving your ai file save it as an older ai format. Or save it as a pdf/x file which should preserve the vector information and Corel Draw can import that too. Also .wmf, .svg, or .eps formats are supported by both. If they still insist that it must be cdr. I guess you could download a trial of Corel Draw and make the conversion yourself.

Haha, usually it's the other way around. Most printers I know don't like to accept .cdr files, LOL.

Corel has a plug-in for making converting illustrator files happen a little more seemlessly. However, the transparency functions in Illustrator don't always play well with Corel. And vice versa.

The usual method we use to go from Illy to Corel is to save in Illy as an .eps.
But bad things can happen to your transparency when doing that as well. If transparency interacts with raster images, you get an even bigger mess.

Seriously talk to the printer about a way to hand off your file.
And get a proof.
Then think hard about a printer that doesn't use all of the industry standard softwares. Maybe they are cheaper for a reason.