You absolutely not only have the right to ask for references, but the obligation. The size and length of time in business is often simply organic growth through name recognition, and may not mean that their clients are necessarily happy. I always provide references when requested, and do so quickly – name, title, email and phone – along with a summary of the project. My own opinion, and others may differ, is that this is a red flag. You want your potential vendor to show that they are responsive, not evasive.
While this isn't an ideal suggestion, you may want to open your RFI again or consider another response from the first round. You probably don't want to choose the other vendor solely by default.
Best of luck.
Dori Kelner, MS | Managing Partner
Sleight-of-Hand Studios LLC
email@example.com | 703.758.7178 | http://www.sohstudios.com
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