You can set up assessments in your Blackboard course that support the submission of an audio file. Note it is the same process for Video files.
We have guidance for Setting Up a Submission Point, How Students Submit and Self-Paced Course on Creating Better Audio:
How to set up an Assignment: Setting up a Panopto Submission
How to Submit: Creating and Submitting for Assessment
Self-paced course: “Creating Better Audio“
Most students will have either a laptop or a mobile phone to record audio files. If not then students could take out a loan laptop from the IT Centre in the Library which has a bookable loan system.