What do you call a person who makes a booking himself via an online service, for himself and/or a group of persons including himself?

If you are talking about someone who takes the booking on behalf of someone or an organisation else they would be the agent as in travel agent or booking agent also if they make a booking for someone else they can be referred to as that persons agent or representative as in the phase "my secretary acted as my representative in making this booking" for example.

If you are talking about someone who made a booking for themselves only or possibly themselves and immediate family then you would commonly refer to them as the customer.

If you are talking about someone who made the booking on behalf of a group of which they are a part then they would be the primary or organiser. In this case you might say something like "I acted as the primary in making the booking for our book club dinner" or "I was the organiser for our company trip to the theatre" either way it would be understood that you were the one that made the booking for everybody.