Can I run two different white label travel websites under two different brand names?

Yes, this is a normal thing to do, and agencies usually do it when they sell to two very different audiences, for example a leisure brand for holidaymakers and a quieter corporate brand for company travel. Each site can carry its own domain, its own logo and colours, its own content and its own markup, so a visitor on one has no reason to think the other exists.

What you need to decide early is how much the two brands share behind the scenes. Running them as one account with two front ends is simpler and keeps your bookings and customers in a single place, but it also means a markup or a supplier setting changed in one place can affect both unless you set them separately. Keeping them fully apart gives you cleaner separation and cleaner reporting, at the cost of doing most admin work twice.

The part that is not entirely in your hands is the supplier side. Some suppliers are happy for one contract to serve several of your brands, and others expect a separate agreement, separate credentials or separate branding rules for each selling site. What you are allowed to do there depends on the supplier and on the plan you are on, so it is worth confirming before you promise a launch date to anyone.

Write down what the two brands must genuinely have in common and what must stay different, then take that list to whoever sets up your portal, because that conversation is much cheaper before the second site is built than after.