Strictly speaking, if you are the claimant, it’s your case – so it’s your job to prepare the bundles for the hearing and provide copies for the other side and the tribunal. You’ll need 6 copies: 1 for you, 1 for your opponent, 3 for the tribunal, and 1 for the witness table.
In fact, bundles for the employment tribunals are probably more often prepared by the employer’s representatives. The reason for this is that in many cases, only the employer is represented by solicitors. Solicitors have fancy photocopiers and heavyweight hole-punches and people who are practised at preparing bundles, so it tends to be assumed that they should do it because it will be easier for them.
The main advantage in doing the bundles yourself is that it gives you control over when the job gets done – there’s no risk that you will only get your copy the day before the hearing and have a very short time to get to know it, mark it with highlighters, stick post-its on the most important pages etc. But if you trust the solicitors on the other side to do a competent job in reasonably good time, and they are willing to do it, you might as well let them.