“Internet donne DES ailes à un secteur” is the correct form.

Both are wrong.

The correct expression "donner des ailes" can’t be used with the verb "donner" (so "ajouter" doesn’t work), nor you can switch "des" for "les".

Some normal usage cases:

Te savoir heureux me donne des ailes !

Mon entreprise avait du mal à se mettre en marche mais Internet lui a donné des ailes.

The following sentence is grammatically correct:

Internet donne des ailes à un secteur.

But it is still very clumsy since "un secteur" is a vague thing, so the meaning of the sentence is very unclear. However saying:

Internet donne des ailes au secteur de la vente par correspondance.

Would be perfectly valid.