You are having the work printed on a press. I assume lots of business cards. In that case the files will ultimately be converted to CMYK to be printed.
By asking you to do the conversion they're trying to:
1. Save a lot of money (good for you if they charge you less).
2. Dodge responsibility (good for them if they can get away with it).
The CMYK conversion from RGB is correctly done by the press operator. By press operator I mean the operator of that specific press. Not a similar press or another press of the same brand and model, I mean THAT SPECIFIC press. Obviously they don't want to do that job and pay personnel skilled and qualified to do that job. In their defense they'd have to charge you and it's a cut-throat business out there. You can't correctly convert your files to CMYK for this job because you don't have the necessary information. The conversion requires profile characteristics of THE SPECIFIC PRESS. It likewise requires a color-managed and calibrated proofing system.
To make their business model work they just skip the hard part and tell you to send them CMYK PDF files. You're option is to do a generic CMYK conversion and then edit the result as best you can and keep your fingers crossed (all bad ideas).
They do have an FAQ that addresses the topic in which they present false information: Here's the FAQ:
Primo Print | Artwork & Files
And here's the false information:
What color mode should my files be in?
They're showing you in the FAQ what happens if an RGB file is generically converted to CMYK and printed without press operator intervention. It's pretty funny, "These are the same colors, but in CMYK." Obviously they're not the same colors. They should be if they were doing their job right! That's why a skilled press operator is required to do the RGB to CMYK conversion. This is their responsibility dodge: By pressuring you upfront to do a CYMK conversion incorrectly they're dodging what should be their responsibility.
Again in their defense: It's a cut-throat business and do you want to pay more for those business cards to have them do a professional job? What if their price doubled in order for them to do this right?
Good luck,
Joe