First of all, i don’t think it’s illegal as long as you not use those trademarks outside of web browser it self, web browser becomes feature of your game and if that illgial pretty much all web browsers become illegal. Also note that there was games that include web browsers in the game, EVE Online which used extra features to JavaScript scripting to integrate websites with the game, they removed because it was hard for them to maintain. Other is VRChat which had intractable web panels, but they removed it for security concerns. None of them had copyright issues because of that… since they just web browsers.
I think only way to have any copyright issue from web browser in the game is when you explicitly use those trademarked websites in marketing, which i bet you not intend to do.
But if you really worried you can block web browser to specific domain or have entire whitelist, on url change (FOnUrlChanged | Unreal Engine Documentation) you can check if domain is match, if it is block the widget or quickly change address to some other website, showing “Access Blocked by network administrator” or something like that.
Main perpace of web browser widget in UE4 is to include HTML snippets very quickly as some games run news feeds purly on HTML that are included to the game. Some UI systems in software you see are HTML content, because a lot easier to add/edit something on HTTP website on the go then UI of the game or software. Epic store and Steam is good examples of that.