Yes, you can have a mobile app that carries your own name, logo and colours on the Apple and Google stores, so your customers download something that looks like your business and not a generic booking tool. The app runs on the same booking engine as your website, which means the suppliers you have connected, your markups, your cancellation rules and your payment options behave the same way in the app as they do on the desktop site. A booking made in the app lands in your back office alongside the rest, so your staff are not watching two separate systems.
What customers can do in the app is broadly what they can do on your site. They can search and book, see their upcoming trips, open their tickets and vouchers, and get push messages for things like payment reminders or a change to a confirmed trip. How much live travel information you can show, such as a flight time change or a gate, depends on whether the supplier or airline feed you are using actually sends that information to us. If it does not, the app can still show the booking exactly as it was confirmed.
The part that is not in our hands is publishing. Apple and Google review every app against their own rules, they can come back asking for changes, and the time they take is theirs rather than ours. You will also need developer accounts in your own company name, because the app belongs to your brand. Which app features are included will depend on the plan you are on.
Write down the few things you most want your customers to do on their phone, then ask us to confirm which of those your current plan covers before you open your store accounts.