Will my white label travel portal show up on Google so customers can find it on their own?

Yes it can, but it is not automatic and it will not happen quickly. A white label portal is a normal website, so search engines can find and list it the same way they list any other travel site. What matters most is that it runs on a domain name you own, that the pages can be read by search engines, and that there is something on them worth listing.

The booking pages themselves are usually not what brings people in from a search. Search results change every time someone looks, so those pages carry very little weight. The pages that do the work are the ones you write yourself, your home page, your pages about the destinations and trips you actually sell, and any guides or articles you add. Most portals let you add and edit that kind of page, although how much control you get over titles and page addresses depends on your plan and how the portal has been set up.

Be realistic about the limits. If your portal sits on an address that is not your own domain, or it opens in a separate window launched from another site, search engines will find it much harder to credit that content to your business. Travel is also very competitive online, so you are unlikely to appear above the big booking sites for broad searches. Smaller and more specific searches, such as a particular kind of tour starting from your own town, are where a small agency can realistically be found.

Check first that your portal is running on your own domain name and that you can edit your own page titles and text, then ask your provider what else can be changed before you spend anything on promoting it.