Is anyone running phone surveys with customers in Qualtrics?

Could you clarify what you mean by surveys on the phone?
If you'd like to send them an invitation via sms/whatsApp that, depending upon your plan, should be available in the distribution section. Otherwise, you can generate personalized links and send them out via any media and all the relevant information will be captured in the survey.
If you'd like to conduct a voice based survey with the same questions, then one option is to go manually call them and get the answers. Alternatively, it's also possible to use an IVR system that's linked to the Qualtrics API at the backend.
Lastly, if you mean changing the format of the survey and conduct the survey in the form of a conversation on SMS/IM app, that is also possible, however, in my experience, it requires rewriting some questions,

So essentially, we would like to collect some vocal feedback from our customers. We want to use the same survey that we are currently using to send via email invitation for online completion. We would like to call a customer to try and obtain their feedback over the phone and enter the data in the survey as we are speaking with them, and if they say they would prefer an email, then we can send them to link to complete it online.
I understand that an anonymous link can be used for this, however, given it's anonymous, it doesn't tie back to the customer's details in the directory.
We are looking for a way where we can take a customer's feedback vocally and still be tied back to the directory.
I hope that makes sense?

Thanks for your reply, could you please clarify how we would generate 'personalised links'?
At the moment, the personalised link is generated when an email invitation is sent out to the customer... I didn't realise you could generate a personalised link without sending an invite. Could you please explain this in more depth?
I really appreciate your help on this!

Another option is to enter the interviewer's email address as recipient instead of the customer. We do this on our end and are still able to send email invitations.
The emails to interviewers include the customer name in the subject and all the other necessary information in the email body through piping in that data. They then enter all the necessary feedback data through the survey link and it's associated back to the customer, and gets pushed back to Salesforce as well.

asanders under the distribution tab you have the ability to generate personal links.
Qualtrics can generate a list of unique, or “personal”, survey links. These links are tied to specific survey recipients, and the names and any other uploaded embedded data will be included with the responses in the results.
When you generate personal links, Qualtrics provides a spreadsheet that contains a row for each recipient including their name and their personal survey link. 
Just be mindful these links are unique to a customer hence cannot be used more than once.
You can reach out to your customer, open the spreadsheet with the personal links, search for the customer, grab the survey link from the "Link" column in the spreadsheet and complete the survey as you speak to them.
https://www.qualtrics.com/support/survey-platform/distributions-module/email-distribution/personal-links/#GeneratingPersonalLinks