There doesn't seem to be any native option to perform search in the default search engine provider set in the browser.
There used to be a external.IsSearchProviderInstalled native JavaScript method[1], but it was deprecated by browsers, most likely to prevent them detecting and recommending one search provider over the other.
Simplest workaround will be to show a dropdown select list or page modal with search engine options, for the user to be able to select their choice.