Can my travel agency software send my customer an invoice with my own company details on it?

Yes, travel agency software will normally produce an invoice for your customer carrying your own company details rather than the supplier's. You set your business name, address, logo and tax registration details once, and the system puts them on every invoice it raises from a booking, together with the traveller's name, the travel dates and what was sold.

How far you can change the look of that invoice varies from one system to another. Most let you edit the wording, add your terms and conditions, and choose whether the customer sees a single total or a breakdown of the fare and your service fee. Some give you a fully custom template, while others give you a fixed design with your branding dropped into it. If your accountant needs the invoice laid out in a particular way, check that before you commit, because it is not always something you can change for yourself afterwards.

Tax is the part that depends most on where you trade. Software built mainly for one country may apply that country's tax rules by default, and handling the rules of another country may depend on your plan or on extra setup. Ask plainly whether your local rules are supported and what happens when you sell to a customer in another country, rather than assuming it is all covered.

The most useful thing you can do is ask for a sample invoice raised from a test booking with your own details in place, and show it to whoever keeps your accounts before you sign anything.