Can I put the booking search on my existing website instead of building a whole new one?

Yes, in most cases you can keep the website you already have and add the booking search to it, rather than starting again from scratch. Agencies who have spent years building up their pages and their traffic usually prefer this, and there is no reason to throw that away just to start selling online.

There are two common ways it is done. Either a search panel sits on one of your existing pages, so a visitor types in their dates and travellers right there, or your menu carries a link that opens the booking section on its own address under your own name. In both cases the results, the passenger details and the payment pages carry your logo and your colours, so the visitor feels they never left you. Which of the two suits you depends mostly on how your current site is built and how much of it you want to leave untouched.

Be aware of a few limits. Some website builders do not allow you to add anything beyond their own blocks, and in that case the linked version is the practical choice. The styling can be matched closely to your site, but it will not always be identical down to every font and spacing. What you can change also depends on the plan you are on.

The useful thing to check first is what your current website is built on and whether the person who looks after it can add a small block to a page, because that one answer decides which of the two routes is open to you.