In order for a survey to be sent to a customer, it would normally need to be with a trigger or automation via email.
However, if the conversation is via a more ephemeral channel and you don't have the customer's email address on record you're still able to send the survey out manually via a special link.
If during your chat/SMS/other channel session the user provides their email address then the survey can be send out via regular means as documented in the main setup article here.
Allowing agents to manually sent out a link to the survey.
To make it as easy as possible for an agent to include a link to the survey manually, it would be best to setup a Zendesk macro that they can call on at any time.
First, you will need to copy the special link that you will need to paste into the macro.
To access the special link click on the survey icon in the main left bar, then under setup you'll find it at the bottom like this:
Once you've copied the link, you can create a macro by clicking on the cog on the left, then under Macros clicking Add macro on the right of the screen like this:
Now give the macro a name (eg. "Ask the customer to fill survey").
Then under Actions type in the text that asking the requester to do this.
For example "We'd love to hear what you think of our customer service. Please take a moment to answer a couple of questions on how we did by clicking this link".
Then append the link you copied from the setup screen to the text "Clicking this link" like this:
Finally, you can Save and then click to Create the Macro at the bottom right of the screen.
Now your agents will be able to use this macro at any point they choose while talking to a customer via any channel.
Note that the link that gets sent will be clickable by anyone so ensure it is only sent on private channels.