I know very little about databases, but can anyone recommend any simple online (or offline) suitable software please, or offer any advice?

I only require the records to hold this info, Certificate Number, Date, Surname, First Name.

You could use a cloud based spreadsheet like Google.
https://www.google.com/sheets/about/

You can see the data via desktopm or app (android or apple)

What’s the likely volume of data?

If it’s relatively small (upto hundreds of rows) then a flat file with comma separation would do the job. Just read the whole thing every time you need access.

The number of rows would be in the 1000’s within the first year.

The number of rows would be in the 1000’s within the first year.

Just 1,000’s (ie less than 10,000?) Google sheets could be an easy solution

Yes, under 10,000.