You don't have to book through the airlines' websites to qualify for Avios and TPs. And if you book through an agent's website instead, if the agent takes your number and passes it on properly to the airline, then you don't need to give the number directly to the airline.

However, if you do give the number to the agent, it's always worth checking with the airline to make sure that it has been passed on properly. Alternatively, you can usually give the number directly to the airline yourself if you can find your booking on the airline's website or if a phone agent can find the booking for you.

Nevertheless, if you book via an agent, you are not necessarily guaranteed to be booking a ticket of a type that will qualify for Avios and TPs. Some agents sell tickets that do not earn, and (if you're buying a ticket for travel on another airline but want to credit to BA) some tickets for travel on another airline do not earn in the BA scheme. So you would have to be very careful to find out whether or not you will. Unfortunately, in some cases that involves trusting the agent to tell you the truth, and sometimes people are disappointed when they find out that the agent has not told the truth.