You may or may not have a HR department, it may or may not be you. If it is you, how do you find training records for yourself or team members, do you keep them at all?
Keep a folder (I am big on folders) for each person (I recommend the manila spring files, you can recycle them) use the files to keep copies of certificates, training records and any other relevant training information for team members. An employee can take the folder with them to their training session (or the sheet, your call).
Create an A4 form (keep it simple) to enter details that cover off, date, name, training provided, accreditation achieved (or not), signed by attendee and instructor. There can be no dispute then over who attended and who didn’t and of course this information can be used in performance appraisals by both team member and line manager. This process is great if you are conducting regular mandatory trainings on OH & S for example.