If a student does not have an institutional account, he cannot link his eduID to the educational institution. Instead, the student links his eduID to iDIN or eIDAS. You can read more about that here: Make a Backpack
In this case you can issue the edubadge via the e-mail address.
Go to the badge class. To issue an edubadge via direct award, click on Award edubadge(s).

Then check the Enable awarding badges solely based on email

Then the screen will look like this:

Now you can also issue this edubadge based on the private e-mail address.
This private e-mail address must match the e-mail address the user used to create their eduID.
You can also add metadata, such as a number, a personal message or a reference to a URL. To do this, click on Add Metadata.

Be aware that the student has the option to hide the grade when he shares the edubadge.
With the + Add another button you can add the next student.
When you're done, click Send.
When the student has created his eduID with the same e-mail addres where the edubdage is issued on, the edubadge will appear in his/her backpack.