Can I put my own travel app on the App Store and Play Store under my own company name?

Yes, your travel app can sit in the App Store and the Play Store under your own company name, with your own logo, your own description and your own screenshots, so a customer searching for you finds your brand and not the name of the company that built the app.

In practice the developer accounts for both stores are registered in your business name and you stay the owner of them. The app is connected to the same booking engine your website uses, so the flights, hotels and packages a customer sees on the phone are the same ones your team works with every day. When you change a markup or add a supplier, the app follows the same settings instead of needing the work done twice.

It is fair to be honest about the part that is in nobody's hands but the stores themselves. Apple and Google review every app before it goes live, and again for most updates, and they can ask for changes or refuse a listing, most often when an app does little more than open a website inside a frame. That review takes as long as it takes, so a launch date is not something you can safely promise a client. Which features your app carries, and how much of the look and wording you can change yourself, will also depend on your plan and on what each supplier allows through their API.

Before you fix a launch date, ask to see exactly which features your plan includes on mobile and who will hold the store accounts, and make sure both developer accounts are opened in your own company name from the very start.