Yes, it can show up in search results, but it will not happen on its own just because the website is live. A white label site runs on your own domain and in your own brand name, so search engines treat it as your business rather than ours. What decides whether people find you is the content on your pages, how long your domain has been around, and how much competition there is for the searches you care about.
The parts of the site that normally get found are your fixed pages, such as your home page, your destination and tour package pages, your blog posts and your contact page. These are the pages where you can add your own wording, your own photos and the page titles and short descriptions that appear in search listings. Live search and availability pages behave differently, because the results change every time somebody searches, so these are generally not the pages that bring in traffic from Google.
Being honest about the limits, nobody can promise you a position in search results, and anyone who does is guessing. How much control you have over page content, titles and descriptions also depends on the plan you are on, so that is worth confirming before you build your content plan. Paid ads and social media are usually what bring the first visitors while your pages slowly gain ground.
Before you go live, check that the site uses your own domain, that you can edit the page text and titles yourself, and then plan a steady set of pages about the destinations and trips you actually sell, because that is what search engines have to work with.